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UBC Theses and Dissertations

Hausdorff measures in topological spaces Willmott, Richard C. 1965

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The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of RICHARD CURTIS WILLMOTT BoA.(Hon.Math.), Swarthmore College M.Sc.(Elect.Eng.)a Princeton University FRIDAY, JULY 16, 1965, AT 9:30 A.M. IN ROOM 229, MATHEMATICS BUILDING COMMITTEE IN CHARGE CHAIRMAN: G. M. Volkoff M. Anvari D. C, Murdoch D. Bressler M. Sion P. Bullen R. Westwick External Examiner: C. A. Rogers University College London Gower Street WC1, London HAUSDORFF MEASURES ON TOPOLOGICAL SPACES ABSTRACT Given a non-negative set f u n c t i o n t on a family CL of subsets of a metric space X, an outer measure V can be generated on X as f o l l o w s ; f o r B C X and 6> 0, V,B = i n f { 2 z A i B c U A- and f o r i£W, 6 ( ieco lew A ^ d and d i a m A ^ d } and VB = l i m VrB. The Hausdorff s-dimensional and h-measures are s p e c i a l cases of t h i s measure. A number of processes have been suggested f o r generating a measure on an a r b i -t r a r y t o p o l o g i c a l space, which g e n e r a l i z e t h i s Hausdorff measure process i n a metric space. In t h i s t h e s i s we introduce and study a process f o r generating a measure on an a r b i t r a r y space, which a b s t r a c t s the e s s e n t i a l idea behind a l l the Hausdorff measures and t h e i r g e n e r a l i z a t i o n s , and contains them as s p e c i a l cases. In chapter I the concept of a measure generated on a space by a gauge and a f i l t e r b a s e i s introduced. We show that w i t h any such f i l t e r b a s e i s a u t o m a t i c a l l y a ssociated a topology f o r the space, the f i l t e r b a s e top-ology. We then impose d i f f e r e n t c o n d i t i o n s on the f i l t e r -base and deduce r e s u l t i n g p r o p e r t i e s of the f i l t e r b a s e topology and of the measure. M e a s u r a b i l i t y and approxi-mation p r o p e r t i e s of the measure are obtained f o r sets defined i n terms of the f i l t e r b a s e , and then f o r sets defined i n terms of the f i l t e r b a s e topology, such as clo s e d , compact, e t c . In chapter I I we consider measures generated on a t o p o l o g i c a l space. We show that previous measures are s p e c i a l cases of our measure and that known m e a s u r a b i l i t y and approximation r e s u l t s can be obtained f o r them from our general theory. The r e l a t i o n s h i p between the given topology and the to p o l o g i e s of the f i l t e r b a s e s used to generate the various measures i s examined. A number of a d d i t i o n a l processes f o r generating a measure on a topo-l o g i c a l space are i n v e s t i g a t e d and r e l a t i o n s among the various measures are studied. In chapter I I I we consider several processes for generating measures on a quasi-uniform space, showing that a number of the previously studied measures are i n -cluded. In p a r t i c u l a r , we study the measure generated on a uniform space, and obtain some measurability properties by applying our general theory. In chapter IV we work i n a compact Hausdorff space and generate a measure using the uniformity for the space and the process of the previous chapter. For the f i r s t time, r e s t r i c t i o n s are placed on. the generating set function X . We examine some consequences of t h i s re-s t r i c t i o n and then introduce a p a r t i a l ordering on the family of such functions which generalizes the usual ordering on the h-functions .in Hausdorff h-measure theory. This ordering has been used.in. connection with studies of non- <r - f i n i t e n e s s . We show here that i t s i n t e r e s t i s e s s e n t i a l l y l i m i t e d to the metric case. • GRADUATE STUDIES F i e l d of Studyt Mathematics Measure theory Point. Set Topology Functional Analysis M. Sion A. Rogers Kobayashi C. Clark Complex Variables Group Theory Z. Melzak R. Ree HAUSDORFF MEASURES I N TOPOLOGICAL SPACES by RICHARD C. WILLMOTT B. A., Swarthmore C o l l e g e , 1952 M. Sc., P r i n c e t o n U n i v e r s i t y , 195^ A THESIS SUBMITTED IN PARTIAL FULFILMENT OF . THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of Mathematics We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA June, 1965 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t , c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r 8, C a n a d a D a t e i i ABSTRACT S u p e r v i s o r : Dr. M. S i o n . G i v e n a n o n - n e g a t i v e s e t f u n c t i o n X on a f a m i l y (X of s u b s e t s o f a m e t r i c space X, an o u t e r measure V can be g e n e r a t e d on X as f o l l o w s : f o r B C X and 6 > 0, V6B = i n f f J ] r A ± : B C \ J A ± and f o r lEo, iew lew Aj_6 d and diam ^ 6 } and v B = l i m v< B. The Hausdorff. s - d i m e n s i o n a l and h-measures a r e s p e c i a l c a s es of t h i s measure. A number of p r o c e s s e s have been suggested f o r g e n e r a t i n g a measure on an a r b i t r a r y topo-l o g i c a l space,-which g e n e r a l i z e t h i s H a u s d o r f f measure p r o c e s s i n a m e t r i c space. I n t h i s t h e s i s we i n t r o d u c e and s t u d y a p r o c e s s f o r g e n e r a t i n g a measure on.an a r b i t r a r y space, w h i c h a b s t r a c t s t h e e s s e n t i a l i d e a b e h i n d a l l the H a u s d o r f f measures and t h e i r g e n e r a l i z a t i o n s , and c o n t a i n s them as s p e c i a l c a s e s . I n c h a p t e r I t h e concept o f a measure g e n e r a t e d on a space by a gauge and a f i l t e r b a s e i s i n t r o d u c e d . We show t h a t w i t h any such f i l t e r b a s e i s a u t o m a t i c a l l y a s s o c i a t e d a t o p o l o g y f o r the space, the f i l t e r b a s e t o p o l o g y . We then Impose d i f f e r e n t c o n d i t i o n s on the f i l t e r b a s e and deduce i i i r e s u l t i n g p r o p e r t i e s o f the f i l t e r b a s e t o p o l o g y and o f t h e measure. M e a s u r a b i l i t y . a n d a p p r o x i m a t i o n p r o p e r t i e s of the measure a r e o b t a i n e d f o r s e t s d e f i n e d i n terms of t h e f i l t e r b a s e , and t h e n f o r s e t s d e f i n e d , i n terms of t h e f i l t e r -base t o p o l o g y , such as c l o s e d , compact, e t c . I n c h a p t e r I I we c o n s i d e r measures g e n e r a t e d on a t o p o l o g i c a l space. We show t h a t p r e v i o u s measures a r e s p e c i a l c ases o f our measure and t h a t known m e a s u r a b i l i t y and a p p r o x i m a t i o n r e s u l t s can be o b t a i n e d f o r . t h e m from our g e n e r a l t h e o r y . The r e l a t i o n s h i p between th e g i v e n t o p o l o g y and the t o p o l o g i e s of t h e f i l t e r b a s e s used t o g e n e r a t e the v a r i o u s measures i s examined. A number of a d d i t i o n a l p r o -c e s s e s f o r g e n e r a t i n g a measure on a t o p o l o g i c a l space a r e i n v e s t i g a t e d and r e l a t i o n s among.the v a r i o u s measures a r e s t u d i e d . I n c h a p t e r I I I we c o n s i d e r s e v e r a l p r o c e s s e s f o r g e n e r a t i n g measures on a q u a s i - u n i f o r m space,, showing t h a t , a number of t h e p r e v i o u s l y s t u d i e d measures a r e i n c l u d e d . I n p a r t i c u l a r , we s t u d y t h e measure g e n e r a t e d on a u n i f o r m space, and o b t a i n some m e a s u r a b i l i t y p r o p e r t i e s by a p p l y i n g our g e n e r a l t h e o r y . I n c h a p t e r IV" we work i n a compact H a u s d o r f f space and g e n e r a t e a measure u s i n g the u n i f o r m i t y f o r t h e space and t h e p r o c e s s o f t h e p r e v i o u s c h a p t e r . F o r t h e f i r s t t i m e , r e s t r i c t i o n s a r e p l a c e d on t h e g e n e r a t i n g s e t f u n c t i o n x. We examine some consequences of t h i s r e s t r i c t i o n and then i v i n t r o d u c e a p a r t i a l o r d e r i n g on the f a m i l y of such f u n c t i o n s which g e n e r a l i z e s the u s u a l o r d e r i n g on the h - f u n c t i o n s i n Hausdorff h-measure theory. T h i s o r d e r i n g has been used i n connection w i t h s t u d i e s of non-<r- f i n i t e n e s s . We show here t h a t i t s i n t e r e s t i s e s s e n t i a l l y l i m i t e d t o the me t r i c case. V CONTENTS Page INTRODUCTION 1 CHAPTER 0. PRELIMINARIES . . . . . . . . . . . . . . . . 5 1. Set t h e o r e t i c d e f i n i t i o n s and n o t a t i o n . . . . . 5 2. G e n e r a l t o p o l o g i c a l c o n c e p t s . . . . . . . . . . 6 3. Q u a s i - u n i f o r m i t i e s and u n i f o r m i t i e s 8 4. Measure t h e o r e t i c c o n c e p t s . 13 CHAPTER I . THE MEASURE GENERATED BY A GAUGE AND A FILTERBASE . . . . . . . 17 5 . The measure 17 6 . The f i l t e r b a s e t o p o l o g y . . . . . . . 18 7. C o n d i t i o n s on a f i l t e r b a s e I n X . . 20 8. P r o p e r t i e s o f t h e t o p o l o g y 21 9. M e a s u r a b i l i t y theorems 26 10. A p p r o x i m a t i o n theorems 38 CHAPTER I I . MEASURES ON TOPOLOGICAL SPACES . . . . . 4 6 11. The measure f i n a m e t r i c space . . . . . . . . .47 12. The measures <f, <fi , and ^ 2 * n a t o p o l o g i c a l space 49 13. The measure X i n a t o p o l o g i c a l space . . . . . . 55 14. R e l a t i o n s between measures, examples . . . ...... . 6l 15. Measures g e n e r a t e d u s i n g n o n - n e g a t i v e f u n c t i o n s . 63 CHAPTER I I I . MEASURES ON QUASI-UNIFORM SPACES . . . . 7^ 16. The measures y., jjt, and jx$ . . . . . . . . . . . 7^ 17. The p r o p e r t i e s o f andjj? . . . . . . . . . . 77 18. M e a s u r a b i l i t y theorems . . . . . . . 87 v i Page CHAPTER IV. MEASURES ON COMPACT HAUSDORFF SPACES . . . . 91 19. P r e l i m i n a r i e s . . . . . . 91 20. The f a m i l y T . 97 21. S e t s o f non-<T- f i n i t e measure . . . . . . . . . . 110 BIBLIOGRAPHY . 1 1 9 v i i ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o thank h i s s u p e r v i s o r , Dr. M. S i o n , f o r s u g g e s t i n g t h e s u b j e c t of t h i s t h e s i s and f o r guidance g i v e n t h r o u g h o u t the a u t h o r ' s g r a d u a t e s t u d i e s and d u r i n g t h e w r i t i n g o f t h i s t h e s i s . The s u p p o r t of t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada t h r o u g h a s c h o l a r s h i p i s a l s o g r a t e f u l l y acknowledged. I n t r o d u c t i o n G i v e n a n o n - n e g a t i v e s e t f u n c t i o n r on a f a m i l y (X o f s u b s e t s of a m e t r i c space X , an o u t e r measure V can be g e n e r a t e d on X as f o l l o w s : f o r B C X and 6 > 0, - J A B = i n f { XI *A± '• B C L J A , and f o r i G W , A,€ (k and ieco lew diam A A 4 6 } , and v B = l i m V i B . P. H a u s d o r f f [6] i n t r o d u c e d t h i s a b s t r a c t measure (a g e n e r a l i z a t i o n of the l i n e a r measure o f C. C a r a t h ^ o d o r y [5,]), and proved a few b a s i c r e s u l t s f o r i t . He c o n s i d e r e d i n some d e t a i l the measures o b t a i n e d when v a r i o u s r e s t r i c t i o n s were p l a c e d on the s e t f u n c t i o n X, i n p a r t i c u l a r when r B = h ( d i a m B) f o r some c o n t i n u o u s i n c r e a s i n g f u n c t i o n h: R + —> R + } w i t h h(0) = 0 and h ( t ) > 0 f o r t > 0 . The measure g e n e r a t e d u s i n g t h i s f u n c t i o n i s c a l l e d the H a u s d o r f f h-measure, and i n the case t h a t h ( t ) = t s , the H a u s d o r f f s - d i m e n s i o n a l measure. I n t h e s e forms i t has been s t u d i e d e x t e n s i v e l y . Two r e c e n t p a p e r s by W. W. B l e d s o e and A. P. Morse [ 2 ] , and by C. A. Rogers and M. S i o n [ 1 2 ] , have suggested p r o -c e s s e s f o r d e f i n i n g a measure on a t o p o l o g i c a l space w h i c h g e n e r a l i z e t h e H a u s d o r f f measure p r o c e s s i n a m e t r i c space. They o b t a i n some ( i n g e n e r a l , d i f f e r e n t ) , m e a s u r a b i l i t y and a p p r o x i m a t i o n r e s u l t s f o r t h e s e measures. 2 In t h i s t h e s i s we i n t r o d u c e a p r o c e s s f o r g e n e r a t i n g a measure on an a r b i t r a r y space, w h i c h a b s t r a c t s t h e e s s e n t i a l Idea b e h i n d a l l of the above H a u s d o r f f measures and g e n e r a l i -z a t i o n s . R e s u l t s a r e o b t a i n e d w h i c h can be s p e c i a l i z e d t o g i v e many of the known r e s u l t s , and w h i c h throw some l i g h t on the r e l a t i o n between measures i n t r o d u c e d b e f o r e . I n a secondary s t u d y , u s i n g some r e s u l t s f rom t h e a b s t r a c t approach, we ex t e n d some s p e c i f i c theorems f i r s t o b t a i n e d i n a m e t r i c space t o a compact H a u s d o r f f space. I n c h a p t e r I we i n t r o d u c e the concept of a measure g e n e r a t e d by a gauge and a f i l t e r b a s e . We show t h a t w i t h any such f i l t e r b a s e i s a u t o m a t i c a l l y a s s o c i a t e d a t o p o l o g y f o r t he space, the f i l t e r b a s e t o p o l o g y , Independent of any e x i s t i n g t o p o l o g y . We then Impose d i f f e r e n t c o n d i t i o n s on the f i l t e r b a s e and deduce r e s u l t i n g p r o p e r t i e s of the f i l t e r b a s e t o p o l o g y and o f the measure. M e a s u r a b i l i t y and a p p r o x i m a t i o n p r o p e r t i e s of the measure a r e f i r s t o b t a i n e d i n terms o f the f i l t e r b a s e . A d d i t i o n a l c o n d i t i o n s on t h e f i l t e r b a s e a r e th e n a p p l i e d t o g i v e r e s u l t s , s t a t e d i n terms o f the f i l t e r b a s e t o p o l o g y , on m e a s u r a b i l i t y of c l o s e d , c l o s e d Jf^ , and compact Jf^ s e t s , and on a p p r o x i -m a t i o n by , 3 ^ , open and c l o s e d s e t s . I n c h a p t e r I I we c o n s i d e r measures g e n e r a t e d on a t o p o l o g i c a l space. We show t h a t t h e H a u s d o r f f measure i n a m e t r i c space and the measures o f B l e d s o e and Morse [ 2 ] , and of Rogers and S i o n [12] a r e encompassed by the g e n e r a l t h e o r y o f c h a p t e r I and t h a t some of t h e m e a s u r a b i l i t y and a p p r o x i -m a tion r e s u l t s can be s p e c i a l i z e d t o y i e l d e x i s t i n g r e s u l t s f o r t h e s e measures. The l a t t e r two of the above measures a r e d e f i n e d i n an a r b i t r a r y t o p o l o g i c a l space; we examine i n each case the r e l a t i o n between the g i v e n t o p o l o g y and the t o p o l o g y a s s o c i a t e d w i t h t h e f i l t e r b a s e used t o g e n e r a t e t h e measure,,and.point out some consequences o f t h e i r e q u a l -i t y o r d i f f e r e n c e . A number of a d d i t i o n a l p r o c e s s e s a r e suggested f o r g e n e r a t i n g a measure on a t o p o l o g i c a l space, some v a r i a t i o n s of p r o c e s s e s a l r e a d y s t u d i e d , and one a d i f f e r e n t approach. Again,, a l l come under the t h e o r y o f c h a p t e r I,,, and r e s u l t s from i t a r e a p p l i e d t o g i v e p r o p e r t i e s o f t h e s e measures. R e l a t i o n s among t h e v a r i o u s measures of t h e c h a p t e r a r e examined. I n c h a p t e r I I I , we c o n s i d e r s e v e r a l p r o c e s s e s f o r g e n e r a t i n g measures on a q u a s i - u n i f o r m space. We show t h a t ' t h e s e measures i n c l u d e a number of tho s e s t u d i e d i n c h a p t e r I I . In p a r t i c u l a r , we s t u d y t h e measure g e n e r a t e d on a u n i f o r m space, and o b t a i n some m e a s u r a b i l i t y . p r o p e r t i e s f o r i t by a p p l y i n g r e s u l t s f r om c h a p t e r I . Chapter IV i s devoted t o an e x a m i n a t i o n of the pos-s i b i l i t y o f e x t e n d i n g some s p e c i f i c r e s u l t s o b t a i n e d i n a compact m e t r i c space by M. S i o n and D. S j e r v e [13] t o a compact H a u s d o r f f space, c o n s i d e r e d as a u n i f o r m space, u s i n g the measure of c h a p t e r I I I . F o r the f i r s t t i m e , r e s t r i c t i o n s a r e put on the g e n e r a t i n g s e t f u n c t i o n x . We examine some consequences of t h i s r e s t r i c t i o n and then i n t r o d u c e a p a r t i a l o r d e r i n g on such f u n c t i o n s x w h i c h g e n e r a l i z e s t h e u s u a l o r d e r i n g i n a m e t r i c space (see Sl.on and S j e r v e [13], s e c t i o n 6 ) . T h i s o r d e r i n g has been used i n H a u s d o r f f h-measure t h e o r y i n c o n n e c t i o n w i t h . s t u d i e s of non-(T- f i n i t e n e s s . We show here (theorem 21.3), t h a t - i t s i n t e r e s t i s e s s e n t i a l l y l i m i t e d t o the m e t r i c c a s e . CHAPTER 0 PRELIMINARIES In c h a p t e r Owe c o l l e c t d e f i n i t i o n s , n o t a t i o n , and known or e l e m e n t a r y r e s u l t s I n s e t t h e o r y , t o p o l o g y , and measure t h e o r y w h i c h w i l l be needed l a t e r . The o n l y new Idea i s t h e concept of p r o p e r t y Q ( 2 . , 2 . 4 ) . 1 . Set t h e o r e t i c d e f i n i t i o n s and n o t a t i o n . . 1 0 denotes t h e empty s e t . . 2 cjj denotes the s e t of n a t u r a l numbers. .3 A — B = [ x : x 6 A and X £ B } . L e t <§ be a f a m i l y of s e t s . Then . 4 TT& = f l A ; AE® .5 <r<B = U A • J AGfi .6 = {A : A = <r(& <~ B f o r some B € 6} ; '7 (0<r = {A : A = B n f o r some sequence B o f s e t s neco i n & } ; . 8 = {A : A = f~ l B n f o r some sequence B o f sets_ new i n <& ) ; . 9 ^ = ( &e \ i &iT= ( 6 4 V .; . 1 0 (6 i s a cov e r o f A i f f A C f f i j . 1 1 (6 i s an 61 - c o v e r of A i f f |B i s a c o v e r of A and <6 C (X ; . 1 2 (X r e f i n e s f l or d i s a, r e f i n e m e n t o f i f f f o r each AGO. , t h e r e e x i s t s BG /B such t h a t ACB;. • .13 & I s a r - f l e l d I f f fi"-Cfi and (B^C & ; and . l 4 B o r e l ( 6 = vr{ d: & 1B a (T-f i e l d and (6 C 61 } I s the s m a l l e s t < r - f i e l d c o n t a i n i n g 6 . .15 I f yD I s a n o n - n e g a t i v e f u n c t i o n on X X X and A G X , then diampA = sup (p(x,y). : x G A , y E A ) i f A / 0 0 , i f A = 0 . .16 9/ i s a f i l t e r b a s e i f f % i s a non-empty f a m i l y of s e t s such t h a t f o r e v e r y MGW and N E % , t h e r e e x i s t s HE % such t h a t 0 / H C M f l N . ?/ i s a f i l t e r b a s e i n X i f f ?/ i s a f i l t e r b a s e and f o r ev e r y HE?/, H i s a f a m i l y of s u b s e t s of X and 0 E H . I f % i s . a f i l t e r b a s e i n X, then i s a s u b f l l t e r b a s e of 7f i f f ?)( i s a f i l t e r b a s e i n : X. and f o r some (X , 7n = { H O a : H E 9f} . .17 (A^jOcED) denotes a n e t . The o r d e r i n g d i r e c t i n g D w i l l be denoted by » . (see K e l l e y [7]> c h a p t e r 2) 2. G e n e r a l t o p o l o g i c a l c o n c e p t s . Most o f our t o p o l o g i c a l c o n c e p t s a r e based c l o s e l y on th o s e of K e l l e y [7] ( h e r e a f t e r r e f e r r e d t o s i m p l y as K e l l e y ) . 2.1 NOTATION. Suppose ( X , & ) • la a t o p o l o g i c a l space and A C X . Then .1 jj of cour s e denotes t h e f a m i l y of open s e t s , 7 w i l l denote the f a m i l y of c l o s e d s e t s , and <£) the f a m i l y of d i f f e r e n c e s of open s e t s , i . e . £ = { A C X : A = G 1 ^ G 2 f o r some Q1, G 2 E Jj } j . 2 A or G1A denotes the c l o s u r e of A ; . 3 A° denotes t h e i n t e r i o r of A ;. and , 4 bdry A = A~A° i s the boundary of A. 2 . 2 DEFINITIONS. C o n d i t i o n s on a t o p o l o g y . .1 I f (6 i s a f a m i l y of s u b s e t s of a space, X, and x £ X , then t h e s t a r a t x o f (6 i s the u n i o n o f t h e members o f (6 t o w h i c h x b e l o n g s . A co v e r (X of X i s a s t a r - r e f i n e m e n t of <6 i f f t h e f a m i l y of s t a r s of (X a t p o i n t s of X i s a r e f i n e m e n t o f & . A t o p o l o g i c a l space i s f u l l y normal I f f f o r each open cov e r & , t h e r e e x i s t s an open co v e r (X w h i c h i s a s t a r -r e f i n e m e n t , of & . ( T u k e y [16]) .2 A f a m i l y of s u b s e t s of X i s p o i n t f i n i t e i f f no p o i n t of X b e l o n g s t o more th a n a f i n i t e number of members of the f a m i l y . A t o p o l o g i c a l space I s metacompact i f f f o r each open c o v e r <fe , t h e r e i s an open c o v e r w h i c h i s a p o i n t f i n i t e r e f i n e m e n t of & . ( K e l l e y , p. 171) . 3 A f a m i l y (X o f s u b s e t s of a t o p o l o g i c a l space i s l o c a l l y f i n i t e i f f each p o i n t of t h e space has a neighborhood w h i c h i n t e r s e c t s o n l y f i n i t e l y many, members of Q. • A t o p o l o g i c a l space i s paracompact i f f i t i s r e g u l a r and f o r each open c o v e r & , t h e r e i s an open c o v e r which, i s a l o c a l l y f i n i t e r e f i n e m e n t o f <& . ( K e l l e y , p. 156) . . 4 A t o p o l o g i c a l space has p r o p e r t y Q, i f f f o r any open c o v e r (X of X, t h e r e e x i s t s an open co v e r (6 r e f i n i n g (X and such t h a t f o r e v e r y x G X , T T { G 6 ( 6 : x £ G } i s open. 8 2.3 R E M A R K S . R e l a t i o n s b e t w e e n c o n d i t i o n s o n a t o p o l o g y . .1 M e t r i c s p a c e s a r e f u l l y n o r m a l . ( T u k e y [16]) .2 A r e g u l a r s p a c e i s p a r a c o m p a c t i f f i t i s f u l l y n o r m a l . ( S t o n e [14]) .3 A p a r a c o m p a c t s p a c e i s m e t a c o m p a c t a n d a m e t a -c o m p a c t s p a c e h a s p r o p e r t y Q . .4 A t o p o l o g i c a l s p a c e m a y h a v e p r o p e r t y Q w i t h o u t b e i n g m e t a c o m p a c t . L e t X = R + , J = {[Q,a) : a > o } . T h e n ( X , Jr) i s a t o p o l o g i c a l s p a c e w h i c h c l e a r l y i s n o t m e t a c o m p a c t . T o s e e t h a t I t h a s p r o p e r t y Q , l e t (X b e a n y o p e n c o v e r o f X . S e t (£ = |[0:,:n) : n £ c j ] . T h e n (6 i s a n o p e n c o v e r o f X w h i c h r e f i n e s (X , a n d f o r a n y x E X , T r { 0 € (B : x G G } = [0> n ) f o r s o m e n G c o . 3. Q u a s i - u n i f o r m i t i e s a n d u n i f o r m i t i e s . We c o n s i d e r n o w c o n c e p t s a s s o c i a t e d w i t h q u a s i -u n i f o r m i t i e s a n d u n i f o r m i t i e s . F o r a f u l l e r e x p o s i t i o n s e e t h e t w o p a p e r s o f W . J . P e r v i n [10,11] a n d c h a p t e r 6 o f K e l l e y . 3.1 D E F I N I T I O N S . .1 A ° B = { ( x , z ) : f o r s o m e y , ( x , y ) E B a n d ( y , z ) G A J , . 2 I f X i s a s p a c e , A = { ( x , x ) : x G X } i s t h e d i a g o n a l o f t h e s p a c e X X X . T h e f o l l o w i n g a r e i m m e d i a t e c o n s e q u e n c e s o f t h e d e f i n i t i o n s . 3.2 L E M M A S .1 (AXA) O (AXA) = A XA. 9 .2 I f A C X X X , t h e n A o A = A o A = A . .3 I f I and J a r e any i n d e x s e t s and A =U A i , B = U B i , 161 j G J then Ao B =U U A i ° B 1 =U U A i ° B 1 • i e l j e J ' j e J i e l .3.3 DEFINITION. I f X i s a space, U a f a m i l y o f s u b s e t s of X X X such t h a t f o r e v e r y U G K, and V G V , .1 A C U, .2 WDU, and W G X X X => W € K , .3 U f l V G V , and .4 t h e r e e x i s t s W G K such t h a t WowCU, then K i s a q u a s i - u n i f o r m i t y f o r X. I f , i n a d d i t i o n t o the above r e q u i r e m e n t s , f o r e v e r y .5 U" 1 = { ( x , y ) : ( y , x ) G u} 6 V, t h e n K i s a u n i f o r m i t y f o r X. .6 (X, 1A)±S a ( q u a s i - ) u n i f o r m space i f f K i s a ( q u a s i - ) ' u n i f o r m i t y f o r X. 3.4 DEFINITIONS. I f U i s an element of a q u a s i - u n i f o r m i t y . .1 U[A] = { y : ( x , y ) G U f o r some x £ A}, .2 U[x] = U [ { x } ] . The f o l l o w i n g lemmas a r e immediate consequences o f the d e f i n i t i o n s . 3.5 LEMMAS. Suppose K i s a q u a s i - u n i f o r m i t y , U€*W , • V € K , and f o r each 1 6 1, Vj_G V . Then 10 .1 f o r each A, U[A] = U [ x ] ; .2 f o r each A, ( U f l V ) [ A ] C U[A] PI V[A] : .3 f o r each A, U[V]A]] = ( U o V ) [ A ] ; and .4 f o r each x (o v ± ) [ x ] = n V i [ x ] . l e i i G I 3 . 6 REMARK. A q u a s i - u n i f o r m i t y U f o r X g e n e r a t e s a t o p o l o g y 7^ on X c o n s i s t i n g of a l l s u b s e t s G o f X such t h a t f o r each x € G , t h e r e e x i s t s U € V. such t h a t U[x] C G. F o r x G X , {U[x] : U G t y } i s a neighborhood system f o r x. (see P e r v i n [11]) 3 . 7-DEFINITION. A t o p o l o g i c a l space ( X , ^ ) . i s ( q u a s i - ) u n i f o r m l z a b l e i f f t h e r e e x i s t s a ( q u a s i - ) u n i f o r m i t y ll f o r X such t h a t 7^ = Jf . 3.8 THEOREM. E v e r y t o p o l o g i c a l space i s q u a s i -u n i f o r m i z a b l e . P r o o f : (see P e r v i n [11]) . L e t (X, Jf ) be a t o p o l o g i c a l space. F o r each GE >£ l e t S G = ( G X G ) U ( ( X ~ G ) X X ) , .1 and l e t (X = {SQ. : G G j ^ } . P e r v i n shows t h a t (X i s a subbase f o r a q u a s i - u n i f o r m i t y Ii f o r X ( h e r e a f t e r r e f e r r e d t o as P e r v i n 1 s q u a s i - u n i f o r m i t y ) , and t h a t = Jj . 11 3.9 REMARK. P e r v i n p o i n t s out t h a t non-comparable q u a s i -u n i f o r m i t i e s f o r t h e same space X may i n d u c e i d e n t i c a i : . t o p o l o g i e s . The same i s t r u e f o r u n i f o r m i t i e s . ( s e e , f o r example, 19-5) 3.10 REMARK. F o r a g i v e n t o p o l o g i c a l space ( X , ^ f ) t h e r e i s a maximal q u a s i - u n i f o r m i t y V such t h a t 1^ = Jr . We t a k e as a subbase f o r H t h e u n i o n of a l l q u a s i - u n i f o r m i t i e s y such t h a t = J) . Then H i s a q u a s i - u n i f o r m i t y by theorem 6.3 of K e l l e y . To see t h a t T-^ = Jf ' Suppose G E ^ and x £ G . Choose U f r o m P e r v i n ' s q u a s i -u n i f o r m i t y such t h a t U [ x ] C G . Then UG'U and so GG 0^ . Now suppose G £ Tj^ and x G G . Then f o r some U € K , U[x] C G. But t h e r e e x i s t s W 6 K , WCU, such t h a t W = n V ± , 1=1 where V i [ x ] i s a neighborhood i n J} of x f o r i = 1, n. Hence by lemma 3.5.^) W[x] = ( ( ^ V ± ) [ x ] = ( ^ V ± [ x ] , and so W[x] i s a neighborhood i n J) of x. S i n c e W[x] C U[x] C G, we c o n c l u d e t h a t G G & . 3.11 REMARKS on u n i f o r m i t i e s . (see c h a p t e r 6 of K e l l e y ) .1 A t o p o l o g i c a l space i s u n i f o r m i z a b l e i f f i t i s c o m p l e t e l y r e g u l a r . .2 There may be non-comparable u n i f o r m i t i e s i n d u c i n g the same t o p o l o g y on a space. .3 I f V i s a u n i f o r m i t y f o r ( X , ^ ), such t h a t \ = J£ , then U E 1 / i m p l i e s t h a t U i s a neighborhood of A i n t h e p r o d u c t t o p o l o g y on X X X . 12 .4 U Is symmetric i f f U ,- U" 1. For any u n i f o r m i t y , there i s a base of open symmetric members of K , and a base of c l o s e d symmetric members of % . 5. Suppose U i s a u n i f o r m i t y , UGH , V G t y , and V i s symmetric. Then f o r any A, i f A X A C U , then V [ A ] X V [ A ] G Vo-Uo V . .6 A f a m i l y <£> of subsets of X X X i s a base f o r some u n i f o r m i t y f o r X i f f a) U€<& => A C U ; b) i f U6 6 , then I T 1 c o n t a i n s a member of & ; c) . i f Ue& , then f o r some VS (6 , V o V C U j and d) the i n t e r s e c t i o n of two members of (6 c o n t a i n s a member. . 7 A u n i f o r m i t y ZL i s c h a r a c t e r i z e d by the gage of U , i . e . the f a m i l y of pseudo-metrics on X which are u n i f o r m l y continuous on X X X r e l a t i v e t o the product u n i f o r m i t y d e r i v e d from 1A . . 8 For a given completely r e g u l a r space ( X , t h e r e i s a maximal u n i f o r m i t y U such t h a t 7^ = Jj - The demonstration i s analogous t o t h a t i n 3.10 f o r q u a s i - u n i f o r m i t i e s (or see K e l l e y , problem 6 G ) . Note t h a t i f a u n i f o r m i t y c o n s i s t s of a l l neighborhoods of A , then i t i s the maximal u n i f o r m i t y by remark 3.11.3. .9 A paraeompact space i s completely r e g u l a r and the maximal u n i f o r m i t y c o n s i s t s of a l l neighborhoods of A . ( K e l l e y , problem 6 L ) . 1 0 I f ( X , V ) i s a u n i f o r m space and (X, ) i s compact, t h e n Ii i s un i q u e and c o n s i s t s o f a l l neighborhoods o f A . . 1 1 I f (X, U) i s a u n i f o r m space and (X, ) i s compact, t h e n each neighborhood o f a compact subset A of X c o n t a i n s a neighborhood of t h e f o r m U[A] f o r some U 6 1/. . 1 2 I f (X, IX) i s a u n i f o r m space and A G X , then t h e c l o s u r e o f A i n t h e u n i f o r m t o p o l o g y , A = Pi U[A] . . 1 3 I f (X,*W) i s a u n i f o r m space and M C X X X , the n the c l o s u r e o f M i n t h e p r o d u c t u n i f o r m t o p o l o g y on X X X , M = H U o H o U . 4 . Measure t h e o r e t i c c o n c e p t s . 4 . 1 DEFINITIONS. .1JX i s an o u t e r measure on X i f f y u i s a f u n c t i o n on the f a m i l y of s u b s e t s of X such t h a t 1 ) y U 0 = 0 , and i i ) 0^yuA< 2 ^ B n w h e n e v e r A c L J B n C X . new n 6 c u As a l l measures d i s c u s s e d i n t h i s t h e s i s w i l l be o u t e r measures we w i l l hence f o r t h drop t h e q u a l i f y i n g word ' o u t e r ' . . 2 F o r JJL a measure on X, a s e t A i s /i-measurable i f f A C X and f o r e v e r y B C X . jmB = yutBDA). + j u ( B ~ A ) . 14 .3 F o r JJ. a measure on X, % ^ = { A C X : A is^ J L - m e a s u r a b l e } . .4 ju|A, t h e r e s t r i c t i o n of JJL t o A , i s t h e f u n c t i o n V h a v i n g the same domain as jx such t h a t f o r e v e r y B i n t h e domain of JJL, V B = JJL[BC\A) . .5 V i s a f i n i t e submeasure o f JJ. i f f f o r some A w i t h jdk<oo t v =yu|A. .6 F o r jx a measure on X, X i s u-<r- f i n i t e , 1 or t r-f i n l t e , i f f t h e r e e x i s t s a sequence A such t h a t X = l._JA n , where f o r ne<o each nGoj, j u A n <co . .7 I f $ i s a f a m i l y of s e t s , X i s a gauge on c& i f f X i s a f u n c t i o n on & U { 0 } t o the extended n o n - n e g a t i v e r e a l l i n e , such t h a t x> 0 = 0. .8 F o r a measure on X, JJL i s a r e g u l a r measure I f f f o r e v e r y A G X , t h e r e e x i s t s B 6 9^ such t h a t A C B and JJA = JJLB. The f o l l o w i n g theorem i s w e l l known. (See, f o r example, c o r o l l a r y 12.1.1 i n Monroe [9].) 4.2 THEOREM. I f JX i s a r e g u l a r measure on'X and A i s an a s c e n d i n g sequence o f s u b s e t s of X, then u ( U A N ) = l i m j u A n . new n->cx> The f o l l o w i n g i s a form o f t h e w e l l known lemma of C a r a t h ^ o d o r y . 4.3 LEMMA. Suppose p. i s a measure on X, and A C X . I f f o r e v e r y £ > 0 and e v e r y T C X such t h a t jxT<oo t h e r e e x i s t s a sequence D o f s u b s e t s of X such t h a t 1) D n + 1 C D n f o r every new; 2) HCnCA ; new 3) /x(THA) < jLi (T r iD n ) + £ f o r every n6cu; and 4) f o r every P C T and n£(j, JJL( ( P r i D n + i ) U ( P - D n ) ) = j u ( P f l D n + i ) + J u ( P - D n ) , then A isjw-measurable. Proof: Let e>0, T C X , pJKoo, B = O c n • W e s h o w new ^ ( T f l A ) + ju(T~A) sjyr + 2e, which.i m p l i e s t h a t A.isju-measurable. We o b t a i n f i r s t 5) There e x i s t s NGw such t h a t j u ( T ~ B ) < yii ( T ~ D N ) ; + 6 . S e t t i n g P = T f l D n we have ^ ( T f l D n ) = jaP >^(PnD n + 2) U ( P - D n + 1 ) ) by l ) = jw(PnD n + 2) + > i ( P ~ D n + 1 ) by 4) = / i ( T n D n + 2 ) + ^ i ( T r i D n ~ D n + i ) by 1) Hence f o r any M G C J , M M E M ( T O D n - D n + 1 ) ^ £ (uCTflDn).- ^ ( T f l D ^ ) ) n=o n=o ' = j U L ( T f | D o ) + ^ ( T f l D;L) - ^ ( T f | D M + 1 ) - jj£ T D Djy[+2 ) < 2^i(TDD 0) < tt> , and M £ j u ( T n D n ~ D n + 1 ) = lim Z M . ( T f | D n - D n + 1 ) n=o M->tx» n=o / < 2 J U ( T D D O ) < 00 16 Choose N€u) so t h a t f;>i(TnD N-D n +i) < 6 . n=N S i n c e ( T O P N - B ) = U ( T D D N - D N + 1 ) by 1 ) , n > N we have j u(TnD N~B) < 6 . But ^LL(T-B) < j u ( T ~ D N ) + y u ( T f l D N - B ) < yx(T~D N). + £ , w h i c h e s t a b l i s h e s 5 ) • Now j u ( T r i A ) + j u l ( T ~ A ) < ^ I ( TOA) ,+ ^JL (T~B) s i n c e B C A , < ja(TnD N+i) + e + ^ ( T ~ D N ) + £ by 3) and 5 ) , = ^ ( T H D - j j + i ) U ( T ~ D N ) ) + 2 6 by 4 ) . 4 JJLH + 2 e . 17 CHAPTER I THE MEASURE GENERATED BY A GAUGE AND A FILTERBASE In t h i s c h a p t e r we s t a r t w i t h an a b s t r a c t space X, a f i l t e r b a s e % i n X (see 1.16) and a gauge t on some f a m i l y GL of s u b s e t s of X such t h a t 0E& (see 4.1.7). From t h e s e we g e n e r a t e a measure and a t o p o l o g y on X, and t h e n i n v e s t i g a t e p r o p e r t i e s of the measure and of the t o p o l o g y . I n p a r t i c u l a r we o b t a i n c o n d i t i o n s under w h i c h c e r t a i n t o p o l o g i c a l s e t s , such as c l o s e d , c l o s e d Jf^ , and compact Jfj s e t s , a r e measurable ( s e c t i o n 9), and a l s o r e s u l t s on the a p p r o x i m a t i o n of a g i v e n s e t f r om above and below by, measurable s e t s or by t o p o l o g i c a l s e t s ( s e c t i o n 1 0 ) . The p r o o f of theorem 9.5 was suggested by the development i n s e c t i o n 2 of B l e d s o e and Morse [ 2 ] ; theorems 10.3 and 10.4 a r e based on theorem 1 and i t s c o r o l l a r y i n Rogers and S i o n [ 1 2 ] ; and t h e p r o o f s of theorems 10.9* 10.10, and 10.11 a r e e s s e n t i a l l y c o n t a i n e d i n t h o s e of theorems 13.5 -13.7 of Monroe [ 9 ] . The t o p o l o g y i t s e l f i s s t u d i e d f i r s t ( s e c t i o n 8) and t h e key r e s u l t , used r e p e a t e d l y l a t e r , i s theorem 8.1.2, w h i c h e s t a b l i s h e s c o n d i t i o n s under w h i c h a c e r t a i n n a t u r a l f a m i l y forms a base f o r the neighborhood system of a p o i n t . From t h i s we d e t e r m i n e when the t o p o l o g y i s r e g u l a r ( 8 . 1 . 4 ) , H a u s d o r f f ( 8 . 1 . 5 ) , o r g e n e r a t e d by a u n i f o r m i t y ( 8 . 2 ) . 5. The measure V. We now i n t r o d u c e the measure g e n e r a t e d on X by t h e f i l t e r b a s e % i n X and t h e gauge % on Oi . We may assume w i t h o u t any l o s s of g e n e r a l i t y t h a t $C<rP£ 5.1 DEFINITION. For H € % and ACX l e t .1 V^**) A = i n f { t : t = 2 t B f o r some countable H B6(B <6 C H f l f l such t h a t A C < r < & } . (note: i n f 0 = oo ) .2 y P ^ ) A = sup V ^ * ' ^ A. HG# I f no ambiguity can a r i s e as a r e s u l t , we w i l l drop one or both s u p e r s c r i p t s on V. 5.2 THEOREM. V i s a measure on X. Proof: i s c o n s t r u c t e d by Method I of Monroe [9] > pp. 90,91, and so, by theorem 11.3 i n Monroe, i s a measure. Since V i s the supremum of such measures, i t i s agai n one. 5.3 REMARK, y i s a set d i r e c t e d . b y i n c l u s i o n , . s o ( 1 / H A , _ H 6 5 ¥ ) i s a net. I t i s an. i n c r e a s i n g net, i . e . H,NE 9/ and HCN i m p l i e s V* H A ^ V J J A, so we have V A = sup V HA = l i m V R A . He# He# 6. The f i l t e r b a s e topology. We now use the f i l t e r b a s e "X i n X to i n t r o d u c e a topology on X, c l o s e l y r e l a t e d to the measure V. 6.1 DEFINITIONS .1 For new, xex, H[x] = {x} U < r{h€H : x&i] . .2 For HG#, ACX, H[A] = j ^ H [ x ] = AU(T{hGH ; hf|A / 0 J . . 3 The 2^-topology, Jf^ ='{GCX : f o r every xGG, there e x i s t s HG^f such t h a t H [ X ] C G } . The s u b s c r i p t may be dropped i f no ambiguity can r e s u l t . 6 . 2 THEOREM. The 5 ¥-topology i s a topology f o r X. Proof: C l e a r l y Jf^ i s c l o s e d under a r b i t r a r y unions. Suppose B , G E j ^ a n d xGBflG. Then there e x i s t . H,NG°H such that H [ x ] C B and N [ x ] C G . Since W i s a f i l t e r b a s e , t here e x i s t s MEW such t h a t M C H f l N . R e f e r r i n g to d e f i n i t i o n 6.1.1 we see M [ x ] C ( H [ x ] f l N [ x ] ) G B f l G , so B D G E * ^ . F i n a l l y , 0 , X 6 ^ . We note that i f f o r a p o i n t xGX there i s H G 2 / such t h a t x^<rH, i . e . no element of H covers x, then {x} i s both open and c l o s e d i n the !Y-topology. Remark. Throughout the remainder of t h i s chapter a l l t o p o l o g i c a l concepts r e f e r t o the 04--topology. The f o l l o w i n g lemmas w i l l be needed l a t e r . 6 . 3 LEMMAS. I f H, Hi, H 2 £ ^ ; f o r each i G I , Aj_CX; and A C X , B C X , then . 1 H t ^ - A i i = y ^ A i i , .2 H i [ H 2 [ A ] ] = J ^ H i [ H 2 [ x ] ] , and . 3 H [ A ] f l B = 0 i f f AHH[B] = 0 . Proof of .1: Let x E ^ - H [ A ± ] . Then xEHU-jJ f o r some i G I . But A j C ^ - A i , whence H[Aj_] C H[.ie^A^, and so H[iWlAi]3iWlH[ Ai]. 20 On t h e o t h e r hand, f o r x ^ H f J L G l A i l => f o r s o m e y G i e i - A i , xGH[y] f o r some 161, t h e r e e x i s t s yGAi such t h a t x6H[.y] f o r some 161, xGH[ Aj_] x G i e ^ H t A i ] . P r o o f of .2 ; H i [ H 2 [ A ] ] = H i [ ^ H 2 [ x ] ] = W A H ! [ H 2 [ X ] ] , by d e f i n i t i o n 6.1 . 2 and. lemma 6.'3.1. P r o o f of .3: By d e f i n i t i o n 6.1 . 2 , H[A]OB = 0 i f f A PIB = 0 and t h e r e e x i s t s no f6H such t h a t f f l A / 0 and f f | B / 0 i f f A f l H [ B ] = 0 . 7. C o n d i t i o n s on a f i l t e r b a s e i n X. We now i n t r o d u c e c o n d i t i o n s on 0°f w h i c h w i l l a l l o w us t o draw c o n c l u s i o n s about t h e 9^ - t o p o l o g y and a b o u t . p r o p e r t i e s of the measure V . (71) G i v e n . x 6 X and H€W, t h e r e e x i s t H x, H 2 6 ^ such t h a t H i [ H 2 [ x ] ] C H [ x ] . ( 7 H ) . G i v e n H € % , t h e r e e x i s t H i , H 2 6 ? / such t h a t f o r e v e r y xGX, H i [ H 2 [ x ] ] C H [ x ] . (We note t h a t by 6.1 . 2 an e q u i v a l e n t statement would be t h a t , f o r e v e r y A C X , H]_ [ H2 [ A.] ] C H[ A ].) ( 7 H I ) I f A i s c l o s e d , B i s open and A C B , then there e x i s t s H E % such t h a t H [ A ] C B . (7IV) There e x i s t s a sequence H i n It such t h a t f o r every NG!#, there e x i s t s nGw such that H n C N , (7V) Given an open cover of X, there e x i s t s HG9f which r e f i n e s t h i s cover. 7.1 REMARKS. .1 I f 04 s a t i s f i e s (711), then i t s a t i s f i e s (71) . .2 I f % s a t i s f i e s (7V), then i t s a t i s f i e s ( 7 I H ) . Proof: Suppose A i s c l o s e d , B i s open and A C B . Then £ = {B,X~A} i s an open cover of X. By (7V), there e x i s t s H G 9 f which r e f i n e s £ . Now any element of £/ , and hence a l s o of H, which i n t e r s e c t s A i s contained i n B so H [ A ] C B . 8. P r o p e r t i e s of the P / - t o p o l o g y. In t h i s s e c t i o n we deduce p r o p e r t i e s of the topology which r e s u l t from Imposing c o n d i t i o n s (71) and (711) on J¥ . 8.1 THEOREM. Suppose sH s a t i s f i e s (71) . Then . 1 I f HG%, A C X , then there e x i s t s an open G such t h a t A C G C H [ A ] . .2 For x€X, {H[x] ::HG i s a base f o r the neighbor-hood system of x. (H[x] i t s e l f may not be open. See example 8 . 4 . ) . 3 f o r ACX, the c l o s u r e of A, A = Q H [ A ] , and i f f o r some sequence H i n *X , A = H n [ A ] , nGW 22 then A i s c l o s e d . . 4 The 74- -topology i s r e g u l a r . . 5 The % -topology i s Hausdorff I f f D H [ X ] = fx) f o r each xGX. m% Proof of .1: Given xGX and HEO'f, we show there e x i s t s an open set G such t h a t x € G C H [ x ] . Let G = {y€X; f o r some NG%, N [ y ] C H [ x ] } . C l e a r l y G C H [ x ] , Let yGG. Then f o r some N€W,- N[.y]GH[x]. Choose N i , N 2 G % such that N i [ N 2 [ y ] ] C N [ y ] . Then f o r any zGN 2[y], N i [ z ] C N i [ N 2 [ y ] ] C H [ x ] , so N 2 [ y ] C G . Hence G i s open. .2 f o l l o w s immediately from .1 and the d e f i n i t i o n , of the % - t o p o l o g y . Proof of .3: Acf^|H[A] : Given HG3Y, suppose x$H[A]. HG& Then { X ) D H [ A ] = 0 , whence by lemma 6.3.3, H[x]f|A = 0 . By 8.1.1 there e x i s t s a neighborhood of x f r e e of p o i n t s of A and so x^A". We conclude t h a t A C H [ A ] f o r every HE3*. A T D f ^ H t A ] : Suppose x $ A". Then s i n c e X^A i s open, there e x i s t s HGW such t h a t H [ x ] f l A = 0 , by d e f i n i t i o n 6.1.3- Again u s i n g lemma 6.3.3 we have x^H[A]. But H [ A ] D H [ A ] D P l H [ A ] , and hence x ^ f |H[A]. I f f o r some sequence H i n % , A = P l H n [ A ] , new 23 t h e n ACnH[A ] c P|H n[A] = A, K£% neco and A = A. P r o o f o f .4: L e t A be c l o s e d , x ^ A . By d e f i n i t i o n t h e r e i e x i s t s HE % such t h a t H [ x ] f l A = 0 . jChoose H i , H2E9Y such t h a t H i [ H 2 [ x ] ] C H [ x ] . Then H i [ H 2 [ x ] ] fl A = 0 , and so by lemma 6.3.3, H 2 [ x ] f l H i t A ] = 0 . By 8.1.1 t h e r e e x i s t d i s j o i n t open s e t s G 2 and G.]_ such t h a t x G G 2 C H 2 [ x ] and A C G i C % [ A ] . P r o o f of .5: Suppose the 0-f-topology i s Haus.dorff and x€X. F o r any y ^ x, t h e r e e x i s t s HG9f such t h a t y(£H[x]. Hence y ^ f |H[x]. ( T h i s does not use c o n d i t i o n ) Now suppose f |H[x] = {xj f o r each xEX. Then by .3 and Hett .4, t h e % - t o p o l o g y i s T i and r e g u l a r , and hence H a u s d o r f f . 8.2 THEOREM. I f % s a t i s f i e s ( 7 H ) , t h e n t h e r e i s a u n i f o r m i t y f o r X such t h a t the u n i f o r m t o p o l o g y i s the ^ / - t o p o l o g y and hence the ^ - - t o p o l o g y i s c o m p l e t e l y r e g u l a r . P r o o f : L e t M = {{x} : xGX} and s e t U H = <r{hxh : hGHUM) and V = { U H : HE^}. We now check: a) I f UEft, then A C U . 2 4 b) I f USH, then U = U" 1 s i n c e every U H i s symmetric. c) I f UE1A, then there e x i s t s V E <2Z such t h a t VoVCU. Suppose UE*U. Then f o r some H E W , U = < r{hXh : h E H U M J . Choose N]_, N 2 E % such t h a t f o r every x E X , N 1 [ N 2 [ x ] ] C H [ x ] . N o w , : l e t N 6 W , NGN]_n N 2 t o get N [ N [ x ] ] C H [ x ] f o r every x E X , and.set V = cr{f X f : f 6 NUM]. Suppose (x,.y)EVoV. Then f o r some z, (x, z) E V and ( z , y ) E V. By d e f i n i t i o n of V, there e x i s t f i , f 2 E N ( J M such t h a t x , z E f i and z , y E f 2 , whence x £ N [ z ] and zEN[.y]. But { z ) C N [ . y ] i m p l i e s N [ z ] C N [ . N [ y ] ] C H [ y ] and hence xEH["y]. By d e f i n i t i o n then, t h e r e e x i s t s hEHUM such t h a t x,y€h, and so (x,y)EhXh.GU, and VoVCU. d) I f U,VE^, then f o r some WE'W, WC U f l V . Suppose U,V£11. Then there e x i s t H x, H 2 E ^ s u c h that U = <r{ hXh : hEH]_U M} and V =(T{hXh : hEHgUMj. Choose H 3 E % K ^ E 1 ^ E 2 and set W = <r{hXh : h £ H 3 U M f c Now l e t (x,y)EW. Then f o r some h E r ^ U M , (x,y)EhXh. But h EH 3UMC(H 1 r i H 2 ) U M , so hEH]_ U M and h E H 2 U M . Hence (x,y ) E U and (x,y ) E V , so ( x , y ) E U D V . We conclude that WCUnV. By theorem 6 . 2 of K e l l e y , Ii Is a base f o r a u n i f o r m i t y f o r X. We show now th a t the uni f o r m topology Is j u s t the t o p o l o g y . L e t G C X . Then G i s open i n t h e u n i f o r m t o p o l o g y I f f f o r each x£G, t h e r e e x i s t s \J€V. such t h a t U [ x ] C G I f f f o r each x€G, t h e r e e x i s t s U G U such t h a t {y : ( x , y ) € U } C G i f f f o r each xGG, t h e r e e x i s t s HG5V such t h a t {y : (x,y€hXh f o r some hGHUM}CG i f f f o r each x€G, t h e r e e x i s t s E£% such t h a t {y : x,y€h f o r some hGHU M] C G i f f f o r each x€G, t h e r e e x i s t s such t h a t H [ x ] C G i f f G I s open i n the 04 - t o p o l o g y . 8.3 LEMMA. I f 04 s a t i s f i e s (71) and. ( t - I V ) , and A i s c l o s e d , then t h e r e e x i s t s a sequence H i n such t h a t A = O H n [ A ] . new P r o o f : U s i n g (7IV) l e t H b e a sequence i n 04 such t h a t f o r e v e r y N E 9 f , t h e r e e x i s t s n£u> such t h a t rL^CN. Then s i n c e A i s c l o s e d we have by 8 . 1 . 3 A = A = nN[A ] D P|H n[A]DA. m% new 8.4 EXAMPLE. L e t X = R, H r = { {x,y} : |x-y| ^ v] \J [0] , 04 = [ H r : r>0}. F o r AG(T>/- l e t x A = fdiam A i f A / 0 (0 i f A = 0 . Then Of i s a f i l t e r b a s e i n X; the t o p o l o g y i s t h e u s u a l t o p o l o g y ; f o r any x£X, r>0, H r [ x ] = [ x - r , x + r ] , a c l o s e d n e ighborhood of x; 04 s a t i s f i e s t h e f o u r c o n d i t i o n s 2 6 ( 7 1 ) - (7IV) but not ( 7 V ) ; t i s a gauge on <T#; and f o r A C X , V A =( o i f A i s c o u n t a b l e oo i f A i s u n c o u n t a b l e . 8.5 REMARK. Even f o r a g i v e n f i x e d gauge x, t h e f i l t e r -to ^ base n, the measure V and t h e Jr-topology a r e not neces-s a r i l y i n one-to-one c o r r e s p o n d e n c e . We w i l l see l a t e r examples o f i ) d i f f e r e n t f i l t e r b a s e s i n X g i v i n g the,same t o p o l o g y and measure (l4.5)> i i ) d i f f e r e n t f i l t e r b a s e s y i e l d i n g t h e same t o p o l o g y but d i f f e r e n t measures ( l 4 . 6 ) , and i i i ) d i f f e r e n t f i l t e r b a s e s i n d u c i n g d i f f e r e n t t o p o l o g i e s but t h e same measure ( l 4 . 7 ) . 9 . M e a s u r a b i l i t y theorems. The f o l l o w i n g d e f i n i t i o n and lemma a r e t a k e n f r o m a paper by B l e d s o e and Morse [ 2 ] , 9 . 1 DEFINITION. F o r <p a measure on X, A, i s y -compact i f f A C X and g i v e n any £ > 0 , f i n i t e submeasure 8 of <f>, and open c o v e r $ o f A, t h e r e i s a f i n i t e s u b f a m i l y £ o f <& such t h a t 9 A < e ( A n < r £ / ) + e . 9 . 2 LEMMA. A c l o s e d s u b s e t of a compact s e t i s Y-compact. We f i r s t s t a t e two theorems and a c o r o l l a r y on V - m e a s u r a b i l i t y of s e t s c h a r a c t e r i z e d i n terms o f t h e f i l t e r -base 04. 27 9.3 THEOREM. I f f o r some sequence B, A = P l B n , new where f o r each n€o> t h e r e e x i s t s M n + 1 € 5 ^ such t h a t M n + l [ B n + i ] C B n C X , t h e n A i s imme a s u r a b l e . 9.4 COROLLARY. I f 9* . s a t i s f i e s (7H),. A G X , and f o r some sequence H i n 04 , A = P | H n [ A ] , new then A i s V-measurable. 9.5 THEOREM. ' I f 04 s a t i s f i e s (71), A i s v-compact, and f o r some sequence H . i n 04, A.= P | H n [ A ] , new then A i s V - m e a s u r a b l e . We now r e l a t e the r e s t r i c t i o n s on A i n the above theorems t o t o p o l o g i c a l p r o p e r t i e s of A and, u s i n g a d d i t i o n a l c o n d i t i o n s on 04, we o b t a i n a number o f theorems on t h e m e a s u r a b i l i t y o f p u r e l y t o p o l o g i c a l s e t s . 9.6 THEOREM. I f % s a t i s f i e s (71), t h e n compact ^ s e t s a r e V-measurable. 9.7 THEOREM. I f 04 s a t i s f i e s (711) and ( 7 I H ) , t h e n c l o s e d s e t s a r e V-measurable. 9.8 THEOREM. I f 04 s a t i s f i e s (711) and (7IV), t h e n c l o s e d s e t s a r e v - m e a s u r a b l e . 28 9.9 THEOREM. I f 04 s a t i s f i e s ( 7 1 ) and ( 7 V ) , then c l o s e d Jf^ s e t s a r e v - m e a s u r a b l e . 9.10 THEOREM. I f 04 s a t i s f i e s ( 7 1 ) , ( 7 I V ) , and ( 7 V ) , the n c l o s e d s e t s a r e V-measurable. 9 . H REMARKS. We note t h a t i f t h e r e i s any subspace X C X w h i c h i s such t h a t f o r any xGX , t h e r e i s some EGOf such t h a t no element of H c o v e r s x, i . e . x(£<TH> then f o r e v e r y A C X , yA = oo ', and by t h e comment a t t h e end of.-theorem 6.2, Jf^ i s d i s c r e t e on X . Thus the d i s c r e t e t o p o l o g y on X i r e f l e c t s t h e f a c t t h a t a l l s u b s e t s o f X a r e V-measurable. Now i t may happen as a r e s u l t of t h e n a t u r e o f t h e f a m i l y (X t h a t t h e c l a s s of measurable s e t s i s l a r g e r than t h a t g i v e n us by any of t h e theorems 9.6 t o 9.10, u s i n g the f i l t e r b a s e 04' ( F o r example, i f Q. I s t h e f a m i l y o f s i n g l e t o n s , t h e n a l l s u b s e t s of X a r e V-measurable, a r e s u l t w h i c h i s independent of the f i l t e r b a s e 04.) . I n t h i s c a s e , I t may be o f some advantage t o c o n s i d e r t h e s u b f i l t e r b a s e of 04> 0}= { H f i a : H G ^ j . E v i d e n t l y t h e measure i / ^ ' ^ = 1 / ^ ' ^ , but t h e 0}-topology, Jf^ , may be s t r i c t l y l a r g e r than Jfy . I f t h i s i s the cas e , and i f 9? s a t i s f i e s t h e r e q u i s i t e c o n d i t i o n s , we may be a b l e t o a p p l y one o f the theorems 9.6 t o 9.10 w i t h the f i l t e r b a s e 9y t o o b t a i n a s t r o n g e r r e s u l t than t h a t o b t a i n e d u s i n g /¥ . (F o r example, i f i n t h e case above o f (X t h e f a m i l y of s i n g l e t o n s , we f o r m t h e f i l t e r b a s e 9} , then t r i v i a l l y 9] s a t i s f i e s ( 7 H ) and ( T I V ) , and Jf^ i s the d i s c r e t e t o p o l o g y . Then by theorem 9.8, a l l s u b s e t s o f X a r e -V-measurable.) However, 0} may not s a t i s f y enough c o n d i t i o n s t o a l l o w us t o a p p l y any theorems from c h a p t e r I (see example 1 1 . 6 ) , so we cannot a u t o m a t i c a l l y use 0] t o g e t s t r o n g e r m e a s u r a b i l i t y r e s u l t s . A g a i n , i t may happen t h a t a l t h o u g h i t s e l f does not s a t i s f y enough c o n d i t i o n s , a n o t h e r f i l t e r b a s e fy can be found such t h a t i ) 97 I s a s u b f i l t e r b a s e of 7>? , so t h a t V V = V , i i ) I s s t r i c t l y l a r g e r than , and i i i ) / ^ s a t i s f i e s c o n d i t i o n s a l l o w i n g a p p l i c a t i o n , of some theorem g i v i n g , a s t r o n g e r r e s u l t t h a n t h a t o b t a i n e d u s i n g (see example 1 1 . 6 ) U n f o r t u n a t e l y , we know of no g e n e r a l method, In such,a case, of c h o o s i n g a f i l t e r b a s e i n X, optimum i n the sense t h a t u s i n g I t we o b t a i n the l a r g e s t p o s s i b l e c l a s s of measurable s e t s . We note a l s o t h a t the n a t u r e of (X. may r e s u l t i n a l a r g e c l a s s of measurable s e t s a t t h e same time t h a t t h e 0} - t o p o l o g y i s no l a r g e r t h a n the *H-topology, i . e . the 9 ?-topology may not be l a r g e enough t o r e f l e c t t he c l a s s of measurable s e t s . ( F o r I n s t a n c e , i f i n example 8.4 H r c o n s i s t e d of a l l s e t s o f d i a m e t e r < r , w h i l e 67 c o n s i s t e d of a l l d o u b l e t o n s , t h e same measure would be o b t a i n e d , under w h i c h a l l s u b s e t s of X a r e measurable, w h i l e b o t h t h e 04 and t o p o l o g i e s would be t h e u s u a l t o p o l o g y , and t h e b e s t theorem o b t a i n a b l e would be 9.8, g i v i n g . c l o s e d s e t s measurable.) 30 PROOFS 9.12 LEMMA. I f A C X , B C X , and t h e r e e x i s t s KG 04 such t h a t .M[A]TI B = 0 , t h e n . v ( A U B ) = VA + vB. P r o o f : Suppose MG 04, M [ A ] f l B = 0, and 1 / ( A ( J B ) < 0 0 . L e t N G 04, NCM. Then a l s o N [ A ] f | B = 0 . By d e f i n i t i o n 6.1.2 no-he N.can i n t e r s e c t b o t h A and B, so any c o v e r o f AIJB by elements o f N f | ^ can be s e p a r a t e d i n t o d i s j o i n t c o v e r s of A and B. C h e c k i n g 5.1.1 we see t h a t V N ( A U B ) > V N A = V N B. S i n c e V N i s a measure, we have the i n e q u a l i t y t h e o t h e r way a l s o , whence V N ( A U B ) = V NA .+ V N B f o r e v e r y NG 04 such t h a t NCM. Hence by remark 5.3 V ( A I J B ) = l i m V N ( A U B ) = l i m (V NA + V N B ) = l i m V NA + l i m V N B = VA + VB. N€9f NGW P r o o f of 9.3: We use lemma 4.3 w i t h D n = B n f o r each nGw. Let. £>0 and T C X such t h a t v T < 0 0 . To check 4) note M n + l t B n + l l H (X~B n) = 0 f o r each nGw, from w h i c h i t f o l l o w s t h a t f o r each nGoi and P C T , M n + 1 [ P f | B N + 1 ] f| ( P ~ B n ) = 0 . A p p l y i n g lemma 9.12 we o b t a i n V ( ( P f l B n + 1 ) U ( P ~ B n ) ) = v ( P f l B n + 1 ) + V ( F - B n ) f o r a l l nGw and P C T . P r o o f of 9.4: We show f i r s t t h a t A can be put i n the form A = P|N n[A], new where N i s a sequence i n % and f o r each nEto N n+l-[Nn +i[A] C N n [ A ] . We c o n s t r u c t the sequence N by r e c u r s i o n . L e t N 0 = H Q and suppose we have N-^ G f o r i = 1,..., n such t h a t N ±[A] C HjjA] f o r i = 0, ...,n, and i N i + i [ N 1 + 1 [ A ] ] C. % [ A ] f o r i = 1, . . ., n - l . We choose N n + 1 as f o l l o w s : u s i n g (711) choose M G % s u c h t h a t M[.M[A]] C N n[A] . Then choose N n + 1 E % such that N n + 1 C M f l H n + 1 . We have i ) N n + 1 [ A ] C H n + 1 [ A ] , and i i ) N n +i[.N n H-l[A.].] C N n[ A] . Now i ) , and i i ) w i l l be t r u e f o r a l l nGw. Prom i ) we have A c Q N n [ A ] C P)H n[A] = A new new and so A = n NnM-new S e t t i n g B n = N n [ A ] , the c o n c l u s i o n f o l l o w s by a p p l i c a t i o n of theorem 9-3-Proof of 9.5: Let T C X , vT<oo, £ > 0 , and 0 = v|T. We employ lemma 4.3 t o show t h a t A i s V-measurable. Sequences C, D, M, and N are c o n s t r u c t e d by r e c u r s i o n . To s t a r t we set C Q = C i = A; M 0 = M1 = H 0; H 0 D N Q = N]_E ^ ; D Q = X; and D l = M l [ C i ] = H 0 [ A ] . Having obtained C ±, D i , MjG % and % G W 32 s a t i s f y i n g a) C - L i s c l o s e d f o r i = 0,...,n, ( C 0 = A i s c l o s e d by 8.1.3) b) C i + i C C 1 C A f o r i = 0, . . . , n - l , c) QC±_i < QC± + e / 2 1 - ! f o r i = 1, . . .,n, d) b i = M i [ C i ] C H i _ i [ A ] f o r i = 1, . . . ,n,. and e) N-j_[Dj_]GDj__]_ f o r i = l , . . . , n , we c o n s t r u c t C n + i , Mn+1> and N n + i as f o l l o w s : F o r each x 6 C n choose, u s i n g (JI), Exi, H X2, • H xg, and H xij.G 04 such t h a t By a) and lemma 9.2, C n i s V-compact. S i n c e 04 s a t i s f i e s ( 7 1 ) , f o r each x € C n t h e r e i s by 8.1.1 open G X such t h a t Hence { G X : x€C n} i s an open c o v e r of C n and by d e f i n i t i o n 9.1 t h e r e i s a f i n i t e s u b s e t Q C C n such t h a t H x l [ H x 2 [ H x 3 [ H x 4 [ x ] ] ] ] C M n [ x ] C M N [ C n ] C D n . x € G x C H x i | [ x ] . e c n ^ 9 ( c n n I J G X ) + ^ 2 n xeQ and so xeQ n Now s e t C n+1 = nH[C nn U H x 4 ^ ] ] = C l ( C n f l U Hx4 [ x D ^ and choose M n + 1 G % N n + i G % such t h a t Mn+iCCPl^OHn , X G Q N n + 1 C p l H x i , xeQ 33 and s e t D n + 1 = M n + 1 [ C n + 1 ] . We now check: a) C n + i i s c l o s e d . b) . C n + i C C n C A s i n c e C n + 1 C C n = C n C A . c) © C n < 0 C n + 1 + £/2n s i n c e C n + 1 D ( C n n U H x 4 [ x D • xeQ d) D n + i = M n + 1 [ C n + 1 ] G H n [ A ] s i n c e C n + 1 C A , M n + 1 C H n . d) N n + 1 . [ D n + 1 ] C D n : f i r s t , Cn+lCplHCUlWx]] = U P l H f ^ t x ] ] He?* xeQ X 4 xeQ m04 C U Hx3[ Hx4[x3]. xeQ The e q u a l i t y i s o b t a i n e d u s i n g theorem 8.1.3 and the f a c t t h a t t h e c l o s u r e of a f i n i t e u n i o n i s the u n i o n of t h e i n d i v i d u a l c l o s u r e s . Now N n + l [ D n + i ] = N n + 1 [ M n + 1 [ C n + 1 ] ] <= N n + i t M ^ U t j H ^ t H x U t x ] ] ) ] ] xeQ = U > l [ M n + i [ H x 3 [ H x 4 [ x . ] ] ] ] xeQ C U H x l t H x 2 [ H x 3 [ H x 4 t x m ] c : D n -xeQ The second t o . l a s t i n c l u s i o n f o l l o w s from t h e c h o i c e of M n + 1 and N n + i , and t h e e q u a l i t y f rom lemma 6.3.1. The completed sequences s a t i s f y a ) , b ) , c ) , d) and e) f o r each nGw. We now check t h a t t h e sequence D s a t i s f i e s t h e h y p o t h e s i s of lemma 4.3. x ) D n + l C D n b y e ) -2) P ) D n C . P | H n [ A ] = A f o l l o w s f rom d) . new new 3) U s i n g A = C]_, c) and. i n d u c t i o n , , we have 9A4 0 C n + e ( l - 1/2"-1) < 0 C n + & f o r e v e r y n€w, or V ( T f l A ) < v ( T D C n ) + £ f o r e v e r y new. S i n c e C n C D n by d ) , we have f i n a l l y V ( T f l A ) 4 V ( T f | D n ) +e f o r e v e r y n€w. 4) I t f o l l o w s f rom e) t h a t N n [ P r i D n ] n(P-Dn-l) = 0 f o r any P C T and n ^ l . Lemma 9.12 t h e n g i v e s us l X ( P D D n ) U ( P - D n - l ) ) = v ( P r i D n ) •+ V ( P - D n _ i ) f o r e v e r y P C T , and n > l . Pr o o f of 9.6: We show t h a t f o r any compact Jf^ s e t A, t h e r e e x i s t s a sequence H i n 04 such t h a t A =P| Hn[ A]> new where f o r each n£u>, H n + l [ H n + l [ A ^ C H n t A ] -Then s e t t i n g B n = H n [ A ] , we a p p l y theorem 9.3 t o o b t a i n t h e c o n c l u s i o n . Suppose A i s compact,.and A = H G n , new where f o r each n€-w, G n i s open. We assume G 0 = X, s e t H 0 = {X$U<r% and c o n s t r u c t H n r e c u r s i v e l y as f o l l o w s : F o r each xGA, u s i n g (71) choose H n x £ 04 such t h a t H n X [ H n x l - x ^ C G n a n d  H n x [ H n x t H n x [ x " ] C H n _ 1 [ x ] . Now each H n x!;x] c o n t a i n s an open s e t c o n t a i n i n g x by theorem 8.1.1 and s i n c e A i s compact, a f i n i t e number of th e s e open s e t s and hence o f t h e s e t s H n x [ x ] c o v e r s A, i . e . t h e r e e x i s t s f i n i t e Q C A such t h a t ^ x ^ W ^ -Now choose H n C f ^ ) H n x , H n£!¥. Then xeQ, H n f M C H ^ U ^ f x ] ] - ( J H n [ H n x [ x . ] ] xeQ xeQ by lemma 6.3.1. S i n c e H n C H n x f o r each x£Q, S i m i l a r l y , Then H n [ A ] C U H n x [ H n x [ x ] ] C G n . xeQ H n [ H n [ A ] ] c U H n X [ H n x [ H n x [ x ] ] ] C H ^ A ] . xeQ A C n H j A l c O G n = A new new and so A .=PlH n[A]. new Pr o o f o f 9 . 7 : We show t h a t f o r any c l o s e d Jf^ s e t A, t h e r e i s a sequence H i n 04 such t h a t A =OH n[A], new 36 and apply c o r o l l a r y 9.h. Suppose A i s c l o s e d , A = f > m new where G n Is open f o r each n€w. Using ( 7 I H ) , choose H n € % such t h a t H n [ A ] C G n f o r each nGw. Then again AcP|H n[A]cnG n = A new new and A = P | H n [ A ] . new Proof of 9 . 8 : We know by lemma 8 .3 t h a t f o r every c l o s e d s et A, there e x i s t s a sequence H i n 04 such t h a t A = p ) H n [ A ] . new The c o n c l u s i o n f o l l o w s from c o r o l l a r y 9A. Proof of 9 . 9 : i ) By 7 .1 .2 04 s a t i s f i e s ( 7 H I ) . i i ) I f A . i s a c l o s e d ^ s e t , then f o r some sequence H i n 04 , A = P | H n [ A ] . new The proof i s contained i n t h a t of theorem.9.7. i i i ) X i s V-compact. Let T C X , vT<oo, 9= y|T, £>0, and X be an open cover of X. Using (7V), choose N€9tf, N r e f i n i n g £ . Now choose such t h a t VT ^ V M T + £/2. Choose KE04, H C N f l M , so by remark 5 . 3 , a) V T ^ V H T + §/2. Since VtfF<oo, choose countable (6 C H D 6 1 , <S = { B i | lew* such t h a t TQr6 and V H T' ^ < oo. lea) Now choose KGco such t h a t oo £ u B i < <=/2. 1=K+1 Since N i s a refinement of X and HCN, f o r each i ^ K choose GjGJC such t h a t B^C G^ and l e t £ = {Gi : i ^ K j . Now (T~<rg) C c{B± : i>Kj and so V H ( T ~ < r g ) 4 &/2. Hence v H T 4 v H ( T n < r e ) + v H(T - ( r£)4vH ( T n<re ) + £/2 4 V ( T n <r£) + 6/2, and by a ) , VT ^ V(T n <r£) + e . Hence 9 x 4 G(Xfl(?£) + £ and by d e f i n i t i o n 9 . 1 , X i s V - c o m p a c t . The d e s i r e d c o n c l u s i o n now f o l l o w s from i i ) , i i i ) , lemma 9-2 and theorem.9 . 5 . Proof of 9.10: We know from the pr o o f of theorem 9.9 tha t i f 04 s a t i s f i e s (7V), then c l o s e d s e t s are v-compact, and from lemma 8.3: t h a t i f 04 s a t i s f i e s (71) and (7IV), then f o r every c l o s e d set A, there i s a sequence H i n 04 such t h a t A . = f i y A ] . new The c o n c l u s i o n f o l l o w s by a p p l i c a t i o n of theorem 9.5. 10. Approximation theorems. We c o n s i d e r f i r s t s e v e r a l theorems on approximation from o u t s i d e i n which the only r e s t r i c t i o n . o n the set t o be approximated i s th a t I t s measure be f i n i t e . The r e s t r i c t i o n t h a t elements of (X be V-measurable s e t s i s necessary i n a l l the theorems of t h i s s e c t i o n but the f i r s t . 10.1 THEOREM. Suppose 04 s a t i s f i e s (7IV) and A C X . I f f o r every ES04there i s a countable subfamily of Hf)<3 which covers A ( i n p a r t i c u l a r i f VA<<»), then there e x i s t s BGOL^x such t h a t B O A and VB = vA. 10.2 COROLLARY. I f 04 s a t i s f i e s (7IV) and dO'hjv, then V i s a r e g u l a r measure. 10.3 THEOREM. Suppose (X C % y , VA«x>, and E C A. Then g i v e n e>0, the r e e x i s t s B € 6 t r such t h a t E C B and V(Af)B) £ v E + e . 10.4 COROLLARY. Suppose Oi<Z%y, vA<oo, and E C A . Then there e x i s t s D G ^ such t h a t B C D and v ( A f l D ) = VE. 3 9 10.5 COROLLARY. I f X = U A n where f o r each n£w, new A n € % v and VAn<co, and &<Z%y, then v i s a r e g u l a r measure. By p u t t i n g f u r t h e r r e s t r i c t i o n s on the approximated s e t , we can get the f o l l o w i n g r e s u l t s on approximation from i n s i d e . 10.6 THEOREM. Suppose tfC&CTfy,, A<E(<6 rf, (see 1.6) Vi\«x>, E C A., and EG%y. Then given £>0 there e x i s t s C€( (B<r )~ such t h a t C C E and v(E~C)<£. 10.7 THEOREM. Suppose & < Z & C % V , AG((B& f , VA<oo, ECA,.and E 6 \ . Then there e x i s t s C€( (6^ ) ~ such t h a t C C E and v ( E ~ C ) = 0. 10.8 THEOREM. Suppose (X C B o r e l 6 C % v , AGBorel© (see 1.14), and VA<oo. Then f o r each E C A there e x i s t s BGBorel<& such t h a t E C B and vE = vB; and f o r each V-measurable E C A, th e r e e x i s t s CGBorel< 6 such that C C E and V ( E ~ C ) =0. I f i t happens t h a t the s e t s of (X have some t o p o l o g i c a l p r o p e r t i e s and are V-measurable (e.g. the open s e t s i n the c l a s s i c a l Hausdorff measure t h e o r y ) , we o b t a i n i n the above theorems approximating s e t s which a l s o have t o p o l o g i c a l p r o p e r t i e s . I f i n our hypotheses we r e s t r i c t (X to open s e t s , r e q u i r e that open s e t s be v-measurable and put a d d i t i o n a l r e s t r i c t i o n s on 04 and X, we o b t a i n some sharper r e s u l t s . ( R e c a l l t h a t Jf and f denote r e s p e c t i v e l y the f a m i l i e s of open and c l o s e d s e t s . ) 10.9 THEOREM. Suppose tfC^G %v , 04 s a t i s f i e s (71) and (7IV), E G X , vE<oo, and EG%y. Then there e x i s t A G ^ sueh t h a t A D E and v(A~E) = 0, and C S ? f such t h a t O C E a n d v(E~C) = 0. 10.10 COROLLARY. Suppose OL<r~-J)<Z%v, 04 s a t i s f i e s (71) and (7IV), E G X , EG%V, and X. i s <T-finlte. Then the c o n c l u s i o n s of theorem.10.9 s t i l l h o l d . I t i s not the case that the e x i s t e n c e of a Jr^ set c o v e r i n g E G X and. having the same measure i m p l i e s t h a t given £>0, there e x i s t s G G ^ such t h a t G D E and vG<ifE .+£. I t may happen t h a t . a l l non-empty open s e t s have i n f i n i t e measure (as, f o r example, w i t h c o u n t i n g measure on R or on the r a t i o n a l s , and Hausdorff |--dimensional measure on R). To o b t a i n t h i s con-c l u s i o n we need an a d d i t i o n a l r e s t r i c t i o n , on the space. 10.11 THEOREM. Suppose ^ G ^ C ^ , 04 s a t i s f i e s (71) and (7IV), E G X , E G \ , and X = l^jA n,.where f o r each nGw, new vA N <OO and A n G A . Then g i v e n £>0, there e x i s t open G D E such t h a t V ( G ~ E ) < £ and c l o s e d F G E such t h a t - y ( E ~ P) < & . PROOFS Proof of 10.1: Using ( 7 I V ) , choose a sequence H i n 04 such t h a t f o r every NG9^, there e x i s t s nGU) such t h a t H n C N . For each nGw choose countable t^t^CE^OO. such that ACirlBn a n d V (<T(&n) ^ Z! r D < V H A. +: 1/n. *n • De6 n n n 41 Let oo B = P l U ^ r J e a ^ • n=o Then B D A and V H B 4 v H n ( ( r 6 n).4VH n A +, 1/n f o r every n£a>. By remark 5.3, t a k i n g the l i m i t as n->oo g i v e s VB^VA. Since BDA, .we have VB^VA, and so V B = VA. Proof of 10.2: 10.2 i s a d i r e c t consequence of 10.1. Proof of 10.3: Choose H € 9 f such .that VA < V HA + £/2. 'Suppose B C X i s V-measurable. Then v(Af|B) + v ( A ~ B ) = V A ^ V H A + 8/2 4 VH ( A f l B ) + V H(A^B) + 6/2 4 % ( A f l B ) + V(A~B) + £/2. C a n c e l l i n g v(A~B), i n the f i r s t a n d . l a s t e x p r e s s i o n s g i v e s a) v(AflB) ^ yH. ( A f l B ) + 6/2 f o r v-measurable B . Now given ECA,. choose countable (6 C H 00. such that E C ( T ( 6 = B and X ] r D < % E - + £/2. But BGCLf and so i s v-measurable, whence by a ) , v(AflB) 4 V H ( A f l B ) + £ / 2 < V H B . + ^2 4 X ! ^ D + ^ 2 ( s i n c e t6 i s a cover of B ) ^ V H E + £ ^ V E . 4- £ . 10.4 f o l l o w s immediately from 10.3, and.10.5 d i r e c t l y from 10.4. 42 Proof of 10.6: By theorem 10.3 there e x i s t s B G ^ such that A ~ E C B and v(AflB) < v(A~E) + £ . Now A~E. Is v-measurable so iXEfKB) = v((Af|B)~(A~E)) = v(APlB) - V ( A ~ E ) < £ . S e t t i n g C = A~B we have by d e f i n i t i o n 1.6, C £ ( ( 6 ( r ) ~ , and s i n c e E f l B = E ~ C , v ( E ~ C ) < £ . Proof of 10.7: The p r o o f i s i d e n t i c a l t o that, of 10.6 except that the r e s u l t of theorem.10.4 i s used i n s t e a d of that of 10.3. Proof of 10.8: We o b t a i n B from c o r o l l a r y 10.4 and C from theorem. 10.7. Proof of 10.9: i ) Use theorem. 10.1 to choose AGa^C ^ such t h a t A D E and vA = vE. Since E i s v-measurable and yE<oo, .we have v(A~E) = 0 . i i ) We show now t h a t i f B G ^ and vB<oo, there e x i s t s DG \ such t h a t D C B and v(B~D) = 0. By lemma 8.3 and theorem 8.1.1 we have 3>"C Jf^ , so Jfd.'fg. and we may s e t , f o r BG*^ , oo oo B = i H U n=l 1=1 where f o r each . nGw and . IGw, • F(n, i ) G . We may assume that F(n, i + l ) D F(n, i ) f o r each iGco. By c o r o l l a r y 10.2, v i s a r e g u l a r measure, so by theorem 4.2 we have f o r each nGw, oo vB = v(BnU F( n^)) = l i m v ( B f | F ( n , i ) ) . 1=1 i 43 Hence f o r each new there e x i s t s a sequence i n such t h a t f o r each kGw, v ( B ~ [ B D P ( n , i n , ).])= VB - - v ( B f | F ( n , i n k k 2 n s i n c e vB<oo and B n P ( n , i n k ) i s V-measurable. Let 00 p ( k ) = P(n,i n•') f o r each. kGw. Then F ( k ) eT and n=l k F.( k )cO F ( n , i ) f o r every nGto, . i = l whence F ( k ) C B and oo V B - v F ( k ) = y ( B ~ F ( k ) ) = i / ( B ~ H n=l K oo = v(LJ [ B ~ ( B f l F ( n , i n )) ]) by de Morgan's law, 1=1 K CO -i ^ Z — = i A . n=l k 2 n Set oo D = U F ( k ) . k=l Then DE.% , D C B , and v ( B ~ D ) = v B - y D <r 1/k f o r every kGw. Hence v ( B ' - D ) = 0 . Now l e t EG % v> vE<oo. Using i ) , choose B G ^ 4 such t h a t B D E and v B = v E . Since i ; ( B ~ E ) = 0, we choose QG«^j such that Q D B ~ E and vQ = 0. Using i i ) , choose D G ? r such t h a t D C B and 1/(B~D) = 0. Now V E = v B = V D = VD - VQ = v ( D ~ Q ) . 4 4 -Set C = D~Q. Then C6 % , C C E , vC = vE and so V ( E ~ C ) .= 0. Proof of 10.10: Let X = LJAn where f o r each new, new vA n<oo. By theorem; 10.1, choose f o r each new, B n 6 such that B n D A n and v B n = v A n . Let E n = E f l B n , so E n £ % v and y E n < oo;. By 10.9 choose f o r each new, C r j e f ^ such t h a t C n C E n and y ( E n ~ C n ) = 0. Set c = U c n . new Then CG%, C C E and new whence oo V ( E ~ C ) 4 2 v ( E n ~ C n ) = 0. , n=o We now apply t h i s r e s u l t t o X ^ E to o b t a i n A 6 ^ such that A D E and i / ( A ~ E ) = 0 . Proof of-10.11: For each new l e t , E n = .EfYA n and, u s i n g •10.9 choose B n e &± such t h a t B n D E n and v(B n — E n ) = 0. L e t B n = O s m > lew 1 where f o r each 160), B ^ E ^ and B n^D ®n ' T h e n 0 0 > v E n = v B n > v ( A n f l B n ) = l i m v.(A -hB n ) . l->oo i Choose i6w such t h a t v ( A n r i B n i ) < V E n .+ 6/2"+! and set G n = A n f l B n 6 Jf. Since E n Is V-measurable and v E n < 6 o , v ( G n - E n ) < £ / 2 n + l . Set G = U G n ^ -neu) Then E C G , Co-Eye U ( G n ~ E n ) , . new and V ( G ~ E ) < X > ( G n ~ E n ) < e . , new To o b t a i n P G T , F C E , v ( E ~ F ) < £ , apply the above r e s u l t t o X--*E. CHAPTER I I MEASURES ON TOPOLOGICAL SPACES In t h i s chapter we s t a r t w i t h a t o p o l o g i c a l space ( X , $ ) and a gauge X on some f a m i l y (X of subsets of X such t h a t 0€.(X. Our aim i s to study measures on X generated by X and J) through processes which are g e n e r a l i z a t i o n s of the w e l l known Hausdorff process i n a m e t r i c space (see Method I I of Monroe [ 9 ] , p. 105)• We f i r s t c o n s i d e r the Hausdorff process i t s e l f , showing th a t the standard r e s u l t s can be obtained by a p p l i c a t i o n of the g e n e r a l theory developed i n chapter I. G e n e r a l i z a t i o n s of the process were i n t r o d u c e d by Bledsoe and Morse [2) and by Rogers and Sion [12]. We show t h a t each of these cases can be obtained as an a p p l i c a t i o n of the theory i n chapter I. More s p e c i f i c a l l y , i n each case we c o n s i d e r a f i l t e r b a s e 5V i n X (04 T) and see t h a t the known measure i s -yv '. Since p r o p e r t i e s {*,x) of V are s t a t e d i n terms of the Of -topology, i t i s important t o study the r e l a t i o n between the given topology and the 04-topology. In p a r t i c u l a r we determine how con-d i t i o n s on ^ a f f e c t the % - t o p o l o g y and i t s r e l a t i o n t o >6, thereby throwing some l i g h t on the r o l e p l a y e d by such con-d i t i o n s . We suggest v a r i a t i o n s of the processes used by Bledsoe and Morse, and Rogers and Sion and determine some r e l a t i o n s between v a r i o u s of these measures. F i n a l l y , an approach from a somewhat d i f f e r e n t p o i n t of view i s i n t r o d u c e d , and shown t o i n c l u d e a l l the measures s t u d i e d p r e v i o u s l y , as w e l l as y i e l d i n g , another measure. 11. The measure f i n a me t r i c space. In t h i s s e c t i o n we suppose t h a t the topology & Is induced by some m e t r i c p on X. A l l m e t r i c concepts r e f e r t o P' The standard m e t r i c measure ]T- generated by a gauge X on a f a m i l y Q_ /(Method I I of Monroe [ 9 ] , p. 105), i s giv e n by 11.1 DEFINITION. For A C X „ and 6>0 X A = l n f { J ] « B i : A c U B i * f o r e a c n IGO) . iGu) and diam B± < o~}. ?A = l i m fs A. <S-H>o To see t h a t the theory of chapter I . a p p l i e s t o T, l e t 11.2 DEFINITIONS. H r = {ACX: diam A ^ r j , 04 = {H r : r> o } , 9) = {HD67 : EE04\. Then % and are f i l t e r b a s e s i n X, 9} i s a s u b f i l t e r -base of 04 , and J = = \^>'c) . The w e l l known proper-t i e s of J* w i l l f o l l o w from the r e s u l t s .of chapter I and the f o l l o w i n g e a s i l y v e r i f i e d , lemmas. 11.3 LEMMAS. .1 The 04 -topology i s the me t r i c topology, i . e . = T . 48:-. 2 94 s a t i s f i e s ( 7 1 1 ) and (7IV). S p e c i f i c a l l y , we have the f o l l o w i n g theorems. 1 1 . 4 THEOREM. I f A i s c l o s e d i n & , then A i s J-measurable. We note that s t r o n g e r m e a s u r a b l l i t y r e s u l t s may be a v a i l a b l e (see remarks 9 i 1 1 and example 1 1 . 6 ) , 1 1 . 5 THEOREMS. Suppose (2C J/ . Then . 1 J i s a r e g u l a r measure. . 2 I f JE<00 and E £ % f , then there e x i s t s D £ ^ such t h a t D D E and ; f ( D ~ E ) = 0 , and C 6 y r such t h a t C C E and J*(E~C) = 0 . . 3 I f X i s J-ir-finite, E£%f, then there e x i s t such t h a t D D E and ;f(D~E).= 0 , and c e y f such t h a t C C E and ;f(E — C ) , = 0 . . 4 I f X = LJ Gn> where f o r each nGa), G nG Jf and new j G n < 00 , E £ *hfy and £ > 0 , then there e x i s t open G D E r such t h a t j f ( G ~ E ) < £ and c l o s e d P C E such t h a t f(E ~ F ) < £ . PROOFS Proof of 1 1 . 4 : Use 1 1 . 3-and theorem 9.8. ' Proof of 1 1 . 5 : . 1 : Use 1 1 . 3 , 1 1 . 4 and c o r o l l a r y 1 0 . 2 . . 2 , . 3 , and . 4 : Use 1 1 . 3 and then r e s p e c t i v e l y theorems 1 0 . 9 , 1 0 . 1 0 , and 1 0 . 1 1 . 1 1 . 6 EXAMPLE. On o b t a i n i n g a . s t r o n g e r m e a s u r a b l l i t y r e s u l t by c h o i c e of an a p p r o p r i a t e f i l t e r b a s e . Let- X = R 2, 49 (X = { A : f o r some x 0 G R, y £ R , and KGw, A = {(x,y) : x = x Q or f o r some n>k, x = x 0 + l / 2 n } }. Let 04, 7\ be d e f i n e d as i n 1 1 . 2 . Then by 9.8, c l o s e d s e t s i n the u s u a l topology are /-measurable. Now 9? does not s a t i s f y ( 7 1 ) , so we cannot get any m e a s u r a b i l i t y r e s u l t s u s i n g , L e t (B = { A : f o r some x 0 £ R and y £ R , A C { ( x , y ) : f o r some s > 0, | x - x Q| < s j- }, and % = { H f l f i : H 6 ^ | . Then 7? i s a s u b f i l t e r base of %, J= , % s a t i s -f i e s ( 7 1 1 ) and (7IV), and so c l o s e d s e t s i n Jf^ are /-measurable. C l e a r l y Jf^ s t r i c t l y c o n t a i n s Jf^ . 1 2 . The measures ^P, *f^> and ^ 2 lR ®. t o p o l o g i c a l space. The measures 9 and ^ 2 helow were i n t r o d u c e d and s t u d i e d by Bledsoe and Morse [ 2 ] and by C. A. Rogers and M. Sion (unpublished) r e s p e c t i v e l y . 1 2 . 1 DEFINITIONS . 1 F a m i l i e s of open covers. c o v e r t = {& : & C £ , <r(B = X, and 0 G (£ } . cover = {(6 : (0 C J ^ i <& i s countable, <T(r) = X, and 0 G 8 } . cover 2 ^ = {<&: &> , (B i s f i n i t e , <T(B = X, and 0 G & } . . 2 For A C X , 16 a cover of X. ^ A = i n f { X! i s a countable refinement of B^S & , , and A C <T 6 } . 50 .3 For A C X , "PA = sup <PfeA. <B 6 cover A (f 1A= sup <ffeA. G c o v e r - ^ f 2 A = sup (6Gcover 2^r To apply the theory of chapter I we set 12.2 DEFINITIONS. H = {A: A C B f o r some BG<&}. 0 4 = {Hfe: & G cover &} . Xi = [H^: (0 G cover!&} . 0 + 2 = ( V £ G c o v e r 2 ^ j . £° = # - t o p o l o g y . = ^ - t o p o l o g y . ^ - t o p o l o g y . Then 0 4 , # 2 are f i l t e r b a s e s The r e l a t i o n s between the g i v e n topology Jb and the induced t o p o l o g i e s til1, J S 2 , and p r o p e r t i e s of the f i l t e r -bases 0 4 , 0+2 are i n d i c a t e d i n the f o l l o w i n g theorem. 12.3 THEOREM. .1 Jf2^^1 CZ 3° <Z Jr. I f i s r e g u l a r , then .2 = Jfl = = 4 , • 3 !W5 #i> # 2 s a t i s f y c o n d i t i o n ( 7 1 ) , and .4 04 s a t i s f i e s c o n d i t i o n ( 7 V ) . In g e n e r a l , Jf ^ j&° as we show i n 1 2 . 6 . On the other hand, r e g u l a r i t y of dt i s not needed f o r JS = & 2 , as we show i n the pr o o f . In view of 8 . 1 . 4 , £ = J) 0 and 04 s a t i s f i e s ( 7 1 ) i f f dt i s r e g u l a r . A p p l y i n g the r e s u l t s of chapter I we then get the f o l l o w i n g m e a s u r a b l l i t y theorems ( a l r e a d y known f o r <P and <P2) • 1 2 . 4 THEOREMS. .1 I f i i s r e g u l a r then c l o s e d Jf^ s e t s are <f-measurable and compact Jf^ s e t s are ^i, <f2-measurable. . 2 I f i f i s normal, then c l o s e d s e t s i n Jj are f, f l > ? 2 _ m e a s u r a b l e . ( S i n c e s i n g l e t o n s are not assumed c l o s e d , n o r m a l i t y doe not imply r e g u l a r i t y . ) Again, s t r o n g e r r e s u l t s may be a v a i l a b l e , as i n d i c a t e d i n the d i s c u s s i o n i n remarks 9 . 1 1 . To o b t a i n approximation r e s u l t s we r e q u i r e t h a t (X(^%a> o r or C2C % w . In any of these cases we can apply d i r e c t l y theorems 1 0 . 3 to 10.8. In g e n e r a l the three measures <P, ¥i, ¥2 a r e d i s t i n c t , as i s shown in, 1 2 . 7 . I t f o l l o w s immediately from the d e f i n -i t i o n s , however, t h a t we always have 1 2 . 5 THEOREM. ^ 2 ^ 1 ^ PROOFS AND EXAMPLES Proof of 1 2 . 3 : . 1 : L e t G G ^ ° and. l e t x £ G. Then f o r some EE 0 4 , H [ x ] C G . But f o r some © G c o v e r t , H = H^.and 52 H[x] = I T { G G 6 : xG G J G ^ . Hence G G ^ , and J)0(zJj. C l e a r l y ^ f e C ^ C / V and so &2 <Z Jl1 (Z Jl° .. .2: We need only show t h a t Jf C Jj2. L e t GSJf and x G G , By r e g u l a r i t y choose c l o s e d C such t h a t x G C C G , and set (& = {G,X~ c } G c o v e r 2 ^ . Then l e t t i n g H = H^ , we have H[x] = G. Thus, f o r each x G G , there e x i s t s H G 9^2 such t h a t H [ x ] C G , i . e . G i s open In the 9 / 2 - t o P ° l o S y ' Note t h a t i n t h i s p roof we need only t h a t C l [ x ] C G . Thus, i f Jj i s a Tj topology, then J) = J)2 . .3: Suppose x G X and HG9V. Then f o r some <& G c o v e r t , H = H^. Now x G G 0 f o r some G QG cfe , and G 0 C H[x] = <r{GG(& : xGG}. By r e g u l a r i t y choose G]_, G 2 G Jr- such t h a t x G G 2 C G 2 C G 1 C ' C G Q and l e t Then and so & ± = {G 0, X ~ G i j G cover 2<# , (0 2 = {Gi, X—G^) G c o v e r 2 ^ , Hi = H ^ G %>> a n d H 2 = H f e gG >/2. H 2 [ x ] = G x, H i t O i ] = G Q, H 2[H![x]] C H[x]. 53 .4: By .2, any cover <& c o n s i s t i n g of s e t s open i n the ^ - t o p o l o g y i s a cover of s e t s open i n &, i . e . (&G c o v e r t , and so H^G IV and r e f i n e s & . , Proof of 12.4: .1: Use 12.3 and. theorems 9.6 and.9.9. .2: The r e s u l t f o l l o w s from theorem 9.3 a f t e r i t has been shown t h a t i f A i s a c l o s e d Jj^ set i n Jj , then there e x i s t s a sequence B of subsets of X such t h a t A = P|Bn, new and f o r each nGo), there e x i s t s N n + 1 £ ^ such t h a t N n + l [ B n + l ] C B n . Suppose A i s c l o s e d i n A and A.= f > n ' neo> where f o r each. nGu, G nG & . The sequences N i n 0+2 a n d B are d e f i n e d r e c u r s i v e l y . To s t a r t , set (& = {G Q, X ~ A ] , N 0 = H<&0, and B Q = G Q. Having obtained B i and N-j_ such that a) Nj_[Bj_] C Bj__]_ f o r 1=0,...,n, (take B_1 = X ) b) . Bj_ i s open f o r i=0,....,n, and c) A C B j _ C Gj_ f o r 1=0, ...,n, we c o n s t r u c t B n + 1 and N n + 1 as f o l l o w s : Let Dn+1 = G n + 1 n B n G i T . Using n o r m a l i t y choose open B n + i such t h a t A C B n + 1 C B - n + 1 C D n + 1 . Let 6 n + l ={Dn+l^ X - B n + 1 } G c o v e r 2 ^ , and N n + j = H ^ ^ E Then the only element of N n + 1 which i n t e r s e c t s B n + 1 i s D n + 1 , so N n + l [ B n + i l = D n + l C B n . For the sequences B and N, a) and c) h o l d f o r every new. c)' assures us t h a t A = H B n -new 12.6 EXAMPLE. J)° does not always c o i n c i d e w i t h & . Let X = R +, j£f = {[0,a) : a > o ). Then f o r any open cover & of X,. and xGX, H 6 [ x ] = X, so Jf° i s the t r i v i a l t o pology. ' 12.7 EXAMPLES. We can have ? 2 / ^1 / .1 A case where «P2 / ^1 • L e t X = R+; Jf = { [ 0 , a ) ; a > o } ; f o r E C X , -CE = | 0 i f E = X or E = 0 [ 1 otherwise . Then f o r any ACX, <& € cover 2«£ , we have Y^A = 0, s i n c e any f i n i t e open cover of X must i n c l u d e X as an element. Hence <P2 i s j u s t the zero measure. On the other hand, f o r any unbounded ACX,, and <&£ c o v e r j j ' such t h a t X ^ c f i , we have f^A = co and so ^ A = 00 . .2 A case where fi / <P. Let X.be any uncountable space w i t h the d i s c r e t e topo-l o g y and r E = 0 f o r any E C X . Then f o r any ACX. and (66 c o v e r ^ , A can be covered by a countable refinement of <f) and so <f^ A = 0 and i s the zero measure. (By 12.5 <P2 i s a l s o the zero measure.) On the other hand, i f A G X i s uncountable and <&> "is the open cover consisting of singletons, then f ^ A = oo and so too VA = oo . 13. The measure A. In a topological space. The measure A 2 defined below was studied by Rogers and Sion [12]. We recall that & is the family of differences of open sets (2.1.1). 13.1 DEFINITIONS. .1 coveriJE) = {(6 : 6 is a countable disjoint cover of X consisting of elements of <£)} . cover2<© = {(&:(& is a f i n i t e disjoint cover of X consisting of elements of <JE)} . .2 For A C X , (& a cover of X, AAA = i n f f T~!tB : £ is a countable refinement of <6 , It <Z(X , and A C i r g J . .3 For A C X , A ] A = sup A^A , ^ecover-^ A 2 A = sup A^A . <feecover2JD The process above breaks down i f we attempt to use arbitrary jD covers. I f the topology is T 1 ? then a cover consisting of singletons would be of this kind and the resulting measure would be in f i n i t e on.any uncountable set, regardless of what gauge "c was used. 56 To apply the theory of chapter I we set 13.2 DEFINITIONS. = {A : A C B f o r some BG<6}. \ = {Hfe: <& G c o v e r ^ j . 7t2 = {H^ : (J3 G cover 2JD} . Jf1 - ^ - t o p o l o g y . = 9/ 2-topology. Then 94and 94^ are f i l t e r b a . s e s i n X (the i n t e r s e c t i o n of two s e t s i n JD i s again a set i n & ), and 13.3 THEOREM. = A 2 . In view of t h i s theorem, s u b s c r i p t s on A w i l l be dropped. 13.4 THEOREMS. .1 For any H G ^ and x £ X , H[H[x]] = H[x], and so 3^, 9i^2 s a t i s f y (7H). .2 JdJ2 = J r l . .3 >$2 i s completely r e g u l a r . .4 I f A i s c l o s e d i n J) , then A i s both open and c l o s e d i n Jf2. .5 If £ i s T]_, then J}2 i s the d i s c r e t e topology. Theorem 13.5.2 f o l l o w i n g was obtained by Rogers and Sion [12]. 13.5 THEOREMS. M e a s u r a b i l i t y . .1 Compact s e t s are A-measurable. 57 .2 I f GG. <t , then G i s A.-measurable. Again, as d i s c u s s e d i n remarks 9.11, stronger r e s u l t s may be a v a i l a b l e . To o b t a i n r e s u l t s on approximation we r e q u i r e t h a t (X C %x . (For example, suppose $ C < 0 . Sets i n JD w i l l be A-measurable by theorem^13.5.2.) In t h i s case we can apply theorems 10.3 t o 10.8. PROOFS Proof of 13.3: Lemma: Given a t o p o l o g i c a l space (X, . I f D - , , . . . , D v r are d i s j o i n t s e t s i n 8 , then N X~{__JlJ4 can be covered by a f i n i t e number of d i s j o i n t i = l s e t s i n <£) . Proof: We note f i r s t t h a t we may assume f o r any set i n £) , D = A ~ B j A, BG £ , t h a t B C A . The proof i s by i n d u c t i o n . n = 1: D]_ = A 1'-'B 1, A x, B-|_G >B . Then X~T>i i s covered by B i and X~A-]_, both elements of <£) . Now suppose t h a t f o r every set D]_,...,Dr or d i s j o i n t r s e t s i n <£) , X ~ j^^D^ can be covered by a f i n i t e number of d i s j o i n t s e t s i n Jb , and l e t D i , . . . , D r + 1 be d i s j o i n t s e t s i n JO . By hy p o t h e s i s we can form the f o l l o w i n g f i n i t e cover of d i s j o i n t s e t s i n JD : •[ D]_, . . .,Dr, C}_, . . ., Cj^). Consider now the f a m i l y Di,...,D r, D r + i = A r + i ~ B r + i , and C j n B r + 1 , C j n ( X ~ A r + 1 ) f o r j = l , . . . , k . i ) I t i s o b v i o u s l y f i n i t e . I i ) I t i s a cover, f o r any p o i n t not i n D]_ to D r + 1 must be i n some Cj. and i n e i t h e r B r + 1 or X ~ A r + 1 < i i i ) I t Is a d i s j o i n t f a m i l y s i n c e the D^, i = l , . . . , r + l are d i s j o i n t , and f o r any j , 1 4 j 4 k, C j n B r + 1 i s d i s j o i n t from Cj f l ( X ~ A r + 1 ) ( s i n c e B r + 1 C A r + 1 ) , and e i t h e r of them Is d i s j o i n t from any Dj_, . i = l , . . . , r , or C m, m/j by hyp o t h e s i s , and from D r + 1 by c o n s t r u c t i o n . i v ) F i n a l l y , i t c o n s i s t s only of s e t s i n <D f o r , s i n c e ( G 1 ~ G 2 ) f l ( G 3 - G 4 ) = ( G 1 f l G 3 ) ~ ( G 2 U G 4 ) , the i n t e r s e c t i o n of two s e t s i n <£) i s again a set i n <D . T h i s completes the proof of the lemma. Now A-^A 2 s i n c e c o v e r ^ D cover 2«© . Ai^A 2 : L e t ACX, a < A i A . Choose (& £ cover-^JD such t h a t A^A > a. Let <£> = . Then 1 J J j € 6 J V = J ^ A H D j ) s i n c e A C ^ J ( A f l D 1 ) and any cover of A which i s ' a r e f i n e -jea J ment of <fe can be broken i n t o d i s j o i n t f a m i l i e s c o v e r i n g the. D^PlA, J E O J . Hence f o r some NEu), J N a< E AjAflDO. j = l 6 J Consider now N S = { D 1 , . . . , D N } U { X ~ U D 1 } . By the lemma above, choose <£ G c o v e r^ J D , <£ r e f i n i n g S ,, and X = ^FA , where F* = D* f o r i = 1,...,N. c x > i = l , ... ,k ± ± Now as i n the case of A A A above, we have K 6 A l s o , f o r 1 i j 4 N, - P j = Dj so A D F j = A f l D j . and A j C A f i P j ) = A 6 ( A n D j ) s i n c e any set contained i n an element of or (B and i n t e r -s e c t i n g F j = Dj i s contained i n P j = D j . Hence N N A . A > X!x»(A P l F ^ = E A f e ( A r i D 1 ) > a . j = l * J j=l® J T h e r e f o r e A 2 A >A^A > a, and so A 2 A > A]_A. Proof of . - 1 3 . 4 . 1 ; Let x G X and H G ^ r For some <&G cover-^JD , H = H^, and f o r some BG(B , x £ B . Now BG<6 i m p l i e s B G H f e so by d e f i n i t i o n 6.1.1 B C H [ x ] . But any element of H c o n t a i n i n g x must be contained i n B, s i n c e H r e f i n e s <B and the elements of <B are d i s j o i n t , so we have H [ x ] C B . Hence H[x] = B and H[.H[x]] = H[B] . But H[B] = B by d e f i n i t i o n 6.1.2 and an argument s i m i l a r t o the one above. Hence H[H[x]] = H[x] f o r each xG X. 60 Proof of 13.4.2: Let and set & = { G , X ~ G } . Then (6 G c o v e r 2 £ and H^G % 2 . F u r t h e r , f o r every xGG, H^tx] = G. Hence G G ^ 2 . C l e a r l y J j 2 (Z J ) 1 . Suppose G G ^ r 1 and x E G. Then f o r some H G c o v e r ^ , H [ x ] C G . But f o r some A ~ B = DGH, ,H[x] = D. Then <£> = { D, B, X ^ A J G c o v e r 2 £ , H^G % 2 , and H^Jx] = D C G . . Hence G E ^ 2 . Proof of 13.4.3: Use 13.4.1 and theorem 8.2. Proof of 13.4.4: A i s c l o s e d i n £ 2 because £ C Jf2. Let (& = {A, X ~ A J . Then <& G cover 2<£), and H^ G # 2. For every x G A, H f i[x] = A. Hence A G Jt2. Proof of 13.4.5: I f & Is T x, then p o i n t s are c l o s e d i n 2 J} and hence by .4 above, open and c l o s e d i n . Proof of 13.5*1: Use 13.4.1 and theorem 9.6. Proof of 13.5.2: Suppose G G ^ and l e t <& = { G , X ~ G } . Then (& G cover2<£) and H^G9f 2. As i n the argument i n the pr o o f of 13.4.1, we show H ^ J G ] = G. The c o n c l u s i o n f o l l o w s from theorem 9.3, s e t t i n g B n = G f o r every nGw. 13.6 EXAMPLE. J}2 may be s t r i c t l y l a r g e r than Jf . Let X = R +, & = { [ 0 , a ) : a> 0 J . Then . i r 2 i s the h a l f - o p e n i n t e r v a l topology, which Is not only l a r g e r than Jf , but l a r g e r than the u s u a l topology on R + as w e l l . (Compare 12.6) 61 1 3 . 7 EXAMPLE. Theorem 1 3 - 5.1 cannot be strengthened to c l o s e d s e t s . I t was shown by Rogers and Slon ( [ 1 2 ] , theorem 8 ) that the measure A d e f i n e d on the r e a l l i n e , w i t h the gauge X on the subsets of R d e f i n e d by rA = diam A i s j u s t the measure f, which i n t h i s case i s known t o be the same as Lebesque measure. But the S¥ 2-topology i n t h i s case i s d i s c r e t e by 1 3 . ^ . 5 and so a l l subsets of R are c l o s e d i f 2 s e t s . 1 4 . R e l a t i o n s between measures, examples. In t h i s s e c t i o n we f i r s t e s t a b l i s h some r e l a t i o n s among some of the measures we have s t u d i e d . We then p r o v i d e examples showing the l a c k of one-to-one correspondence between the f i l t e r b a s e 04, the measure and the 9f-topology mentioned i n remark 8 . 5 . The f o l l o w i n g r e s u l t was obtained by Bledsoe and Morse [ 2 ] , 14.1 THEOREM. I f (X, ^  ) • i s a me t r i c space, then <P =f. 14.2 REMARK. I t was shown by Rogers and Sion ( [ 1 2 ] , theorem 8 , and i n some unpublished work) t h a t i f (X, Jf) i s a separable m e t r i c space, and x i s w e l l behaved i n a c e r t a i n sense, then A= X = f^' I r i t h i s c a s e then (which i n c l u d e s Lebesgue measure and the c l a s s i c a l Hausdorff measures) f = A = f2 = ^1 = V by theorems 12 . 5 and 14.1. 14 . 3 REMARK. In the example i n 12 . 7.2, the space i s me t r i c and i t i s easy t o see th a t CP=T, as i s assured by 62 theorem 14.1. On the other hand, by 12.5, ^ 2 l s the z e r > 0 measure, and s i n c e c o v e r t © = c o v e r 2 ^ , we have a l s o t h a t X= ^2* W e h a v e t h e n f - < P / A - ? ! = <P2-In t h i s sense, *P Is the most s u c c e s s f u l of these measures In g e n e r a l i z i n g from the m e t r i c case. 14.4 REMARK. We note t h a t In example 12.7.1, A i s counting measure, d i f f e r e n t from both ^ and ^ 2 * We now c o n s i d e r examples of d i f f e r e n t f i l t e r b a s e s , measures and t o p o l o g i e s . 14.5 EXAMPLE. D i f f e r e n t f i l t e r b a s e s may y i e l d the same topology and measure. Let X = R; f o r ACX, -cA = diam A. Let Hp = { A C X : diam A ^ r } , M r = {(a,b) : b - a ^ r ] and 04 = { H r : r > 0 } , % = {M r : r > O J . I t i s w e l l known t h a t V = V i s Lebesgue measure, and i t i s c l e a r t h a t = A , ' 14.6 EXAMPLE. D i f f e r e n t f i l t e r b a s e s may g i v e the same topology but d i f f e r e n t measures. In c e r t a i n cases of Hausdorff s-dimensional measures B e s i c o v i t c h [ l ] has shown t h a t u n l i k e the case above, a d i f f e r e n t measure i s obtained i f the c o v e r i n g s e t s are r e s t r i c t e d t o open spheres. Again i t i s c l e a r t h a t the topology i s the u s u a l one. (Another example may be c o n s t r u c t e d e a s i l y from the case i n example 8.4.) 14.7 EXAMPLE. That d i f f e r e n t f i l t e r b a s e s may induce the same measure but d i f f e r e n t t o p o l o g i e s i s j u s t the p o i n t of the remarks i n 9.11. For another example, l e t X = R; f o r A C X , zA = diam A. Then by theorem 8 of [ 1 2 ] , A = X , which In t h i s case i s Lebesgue measure. By 11.3.1 the W - t o p o l o g y a s s o c i a t e d w i t h f i s the u s u a l topology w h i l e by 13.4.5 the 5^-topology a s s o c i a t e d w i t h A i s d i s c r e t e . 15. Measures generated u s i n g non-negative f u n c t i o n s . A weak m e t r i c on X (sometimes c a l l e d a q u a s i - m e t r i c ) i s a non-negative f u n c t i o n p on X X X s a t i s f y i n g p(x,x) = 0 and the t r i a n g l e i n e q u a l i t y p{x,z) 4 p ( x , y ) + p(y,z) f o r x , y , z £ X . Any t o p o l o g i c a l space (X,Jj) has a c a n o n i c a l f a m i l y of weak m e t r i c s | p G : G£ > f t J a s s o c i a t e d w i t h i t (see d e f i n i t i o n . 1 5 . 1 . 1 ) . T h i s f a m i l y i n f a c t generates a q u a s i - u n i f o r m i t y which induces the topology Jf (see P e r v i n [ 1 1 ] ) . I t i s n a t u r a l t h e r e f o r e t o t r y to generate measures u s i n g t h i s f a m i l y and the standard Hausdorff p r o c e s s . In t h i s s e c t i o n we e x p l o r e s e v e r a l avenues along t h i s a p p r o a c h a n d c o n s i d e r measures generated u s i n g non-negative f u n c t i o n s on X XX. We o b t a i n again the measures of s e c t i o n s 12 and 13 and an a d d i t i o n a l one, r\ , which i s very s i m i l a r t o the measure <P. 15.1 DEFINITIONS. .1 For E C X , ^ E ( x , y ) = j l i f x E E and y £ X ~ E [ 0 otherwise. . 64 .2 For <& a f a m i l y of subsets of X, a EGdB ; i j .3 For G a sequence of subsets of X, B Q ( x , y ) - E l / 2 n / b n ( x , y ) . new We note t h a t fa, r ^ , and SQ. are a l l weak m e t r i c s which may f a i l t o be symmetric. In g e n e r a l , r ^ i s not l i k e l y t o be very i n t e r e s t i n g . For example, I f X i s the r e a l l i n e and (B i s the f a m i l y of a l l open i n t e r v a l s w i t h r a t i o n a l end-p o i n t s , then r ^ ( x , y ) = 1 i f f x / y and the measure generated by the Hausdorff process u s i n g s e t s of r^-diameter l e s s than 1 w i l l be i n f i n i t e f o r any uncountable s e t , r e g a r d l e s s of the f u n c t i o n x. On the other hand, i f (6 i s f i n i t e and G i s any o r d e r i n g of the elements of <B then, f o r s u f f i c i e n t l y s m a l l 6>0, dlam r A<6 i f f diam- A < For t h i s reason, we use <& G only S Q t o generate measures. We note a l s o t h a t t a k i n g the infimum r a t h e r than the supremum over a f a m i l y of the weak m e t r i c s does not l e a d t o a n y t h i n g of i n t e r e s t , as i n c r e a s i n g the number of s e t s i n the f a m i l y i n c r e a s e s the f a m i l y of s e t s of zero diameter. In f a c t , i f two s e t s of the f a m i l y (& were d i s j o i n t , then a l l subsets of X would have zero diameter. 15.2 DEFINITION. For any countable f a m i l y <& , l e t G be any o r d e r i n g of the elements of <B , and f o r A G X and 6>0, V A = inf{ S ^ E i : A C l j E i and f o r lew, E±G CL Gf4 lew lew and dianie, E., < . SG 1 yGA = lim % i A, 15.2.1 REMARK. I f D Is any other o r d e r i n g of the elements of <& , we have yG = 'Yp . Hence, we s h a l l w r i t e ^ i n s t e a d of yG. 15.3 DEFINITIONS. For any ACX, t-jA = sup t f i A <&€ cover 3^ %A = sup %A. <BGcover 2«& We then have 15.4 THEOREM. «\>i = y2 = A ' Next,.we c o n s i d e r another f a m i l y of non-negative f u n c t i o n s on X X X which may f a l l t o be weak m e t r i c s . 15.5 DEFINITIONS. .1 For BCX, [ l otherwise .2 For & a f a m i l y of subsets of X, A C X , A = i n f { X ^ E - L : A C U E i a n d f o r 1 € w > E i € # lew i€W and i n f diamd E\ = o | . Be© B i ^A = sup (Be c o v e r t ^ A = sup <&ecover-,i# ^ 2A = sup <&€ cover 2«^ In t h i s case, t a k i n g the supremum r a t h e r than the infimum over a f a m i l y of the f u n c t i o n s ^ does not l e a d t o anyt h i n g I n t e r e s t i n g . I f we have two d i s j o i n t s e t s i n the f a m i l y (S , then and a g a i n we a u t o m a t i c a l l y get i n f i n i t e measure on uncountable s e t s . 15.6 THEOREM. $ = <p , ^ = f-p ^ 2 = <P 2 . In d e f i n i t i o n 15.2, l / 2 n diampu E ± < 6 new J n can be s u b s t i t u t e d f o r d i a r r i o E* <6 (without a f f e c t i n g the G 1 measure ^ = 'Y 2 ) . We have a l r e a d y used the f a c t t h a t the former i m p l i e s the l a t t e r . Although the converse does not n e c e s s a r i l y h o l d , I t i s not hard t o show that the measure obtained u s i n g the f i r s t i s l e s s than or equal t o t h a t obtained u s i n g the second; the p r o o f i s almost i d e n t i c a l t o t h a t i n 15.2.1. In the case of the f u n c t i o n ^g, u s i n g the infimum, t h i s s i t u a t i o n does not h o l d . I f we change the p o s i t i o n where the infimum i s taken, we may get a d i f f e r e n t measure. sup diam<) (x,y) = 1 i f x / y, BG<& B 15.7 DEFINITIONS .1 For <& a f a m i l y of subsets of X, d6(x,y) = i n f ^ U j y ) . .2 For AGX, <B a f a m i l y of subsets of X, o.A = i n f f T>xE, : AcU^, f o r i G w , E.,£ and 6 . ieco 1 lew x diarrid E i = 0}. •7 A = sup K^A ©ecover^ 0 5 K^A = sup rj AA ( g ^ c o v e r ^ Kl2 A = sup i ^ A . <BGcover2^ We compare the ^ -measures and the ^ -measures ( s e c t i o n 1 2 ) . 15.8 THEOREMS. .1 *i4<P, K^Vi, n 2 ^ 2 -.2 I f (X,&) i s f u l l y normal (2.2.1) then r\ = (?. Since m e t r i c spaces are f u l l y normal ( 2 . 3 . 1 ) , n . = ( f I n any me t r i c space,, and hence = "f In any m e t r i c space, by theorem 14.1. Although t h e r e are cases i n which r\ / <f (see example 15.12 below), has many of the same p r o p e r t i e s as «f . To see t h i s we l e t 04^ be the % of 12.2 and c o n s i d e r the f i l t e r b a s e 94^ a s s o c i a t e d w i t h v\. 15.9 DEFINITION. For (6 £ c o v e r t , M* = { A C X : diarr^ A = 0} % l = { M 6 : ^ 6 cover i f J . 15.10 LEMMA. Given x £ X , <B G cover J} , M = M f iG \ . H = H f eG 5^ ,, then H[x] = M[x]. I t f o l l o w s from 15.10 th a t a l l r e s u l t s which depend on the concept H[x] are the same f o r the measures i\ and <P. Thus r e s u l t s analogous t o those i n 12.3 and 12.4 hold a l s o f o r Y\ , and the Wrj-topology i s the same as the ^ - t o p o l o g y . PROOFS Proof of 15.2.1: Let G and D be d i f f e r e n t o r d e r i n g s of the same countable f a m i l y . We show t h a t f o r . E C X , T G E = T D E . L e t 6>0, and choose NGw so that l / 2 N < 6 and £ 1/2* < 6. i=N+l Let = {Gi» . . . ,G^j ] and choose MGco such t h a t {•plf . . , , D M } Z > £ . C l e a r l y M^N. Let £ < l / 2 M . Then f o r E C X , diam s E < £ => sup ( 2 ] l / 2 t J P D (x,y)) <6 D x,y€E j = l J => sup 1/2 J (x,.'y) < £ f o r 3^1 x,yeE J 1/2J diamDj) E < £ f o r j M d i a m / ^ E = 0 f o r j = 1,...,M diamp E = 0 f o r i = 1,...,N. oo J-1 diamp E < 6 Y, 1/21 diamp 1=1 G i diam s E <b, G 69 the l a s t i m p l i c a t i o n f o l l o w i n g from the f a c t t h a t the sum of suprema i s g r e a t e r than or equal t o the supremum of the sums. We have then Y D , e E ^ ^G,6 E> s i n c e by the above r e s u l t , , a n y c o v e r i n g s e t used to d e f i n e the f i r s t can be used f o r the second, and so i n the l a t t e r case the infimum i s taken over a l a r g e r s e t . I t f o l l o w s t h a t s i n c e 6 was a r b i t r a r y . The r e v e r s e i n e q u a l i t y i s proved i n an analogous manner. Proof of 15.4: C l e a r l y % ^^2' W e s h o w °f2 ^A 2 : W e show t h a t g i v e n S E cover 2 <© , there e x i s t s (fe E covev2>£t and 6>0 such t h a t f o r some o r d e r i n g G of <fe , %'6 Suppose it E cover 2£) , £ = {Gj_~ G j _ + n : i = l , , .. . , n j . L e t 6 < l / 2 2 n and (fe = G = { Gi : 1=1, • . . ,2n| E c o v e r 2 ^ r . Now suppose diam s B<6. Then f o r i = l , . . ; ,2n, diamp B = 0 and so B C G 1 i f f B D G i / 0 . Hence f o r some D E 6, DDB. Thus f o r any f a m i l y of elements of s G - d i a m e t e r . l e s s than 6, p r e f i n e s it and we have ^G,6 ^ V A? > f-, : Let 8 e cover-, A , & = G = {G* } , A C X , 1 1 lew and a < Choose 6 > 0 such t h a t %,L A > a ' and N 6 O J such t h a t Let and 1/2N<6 and £ 1/2* < 6. i= N+1 £ = {G 1,...G N, X} G cover 2*#, (6 = { A ~ B : A = TTJC f o r some £ C ^ , ,£ / 0 , and B = <r( P - X ) } . Then <&£cover2<0 f o r 1) <& . i s c l e a r l y f i n i t e , 2) (6 i s a cover of X s i n c e ^ i s a cover of X, 3) f o r any ( A ~ B ) G (& , A and B are both open, and 4) elements of <B are d i s j o i n t . To see t h i s , suppose xG A ~ B f o r some (A<~B)G<r} . Then x i s a member of each element In the i n t e r s e c t i o n forming A and of none of the elements i n the union forming B. T h i s i s c l e a r l y the only decomposition of i n t o two f a m i l i e s f o r which t h i s i s t r u e , i . e . x can be i n no other element of (& . Now suppose D C A ~ B f o r some (A~B)G(fe . Then f o r i = l , . . . , N , D C G ± i f f T>C\G± / 0 and so diamn D = 0 f o r i = l , . . . , N . Hence oo X! 1/2 1 diamp D < 6 i=0 ^O, and so diam,, D < 6. S G Thus f o r any f a m i l y which r e f i n e s <& , every element of th a t f a m i l y has s^-diameter l e s s than 6, whence Since £, was a n . a r b i t r a r y element of c o v e r ; ^ , we then have Proof of 15.6: The c o n c l u s i o n f o l l o w s from the f a c t t h a t i n f diamd B = 0 i f f B C G f o r some GG<6 . GQB G 15.11 LEMMA. For A C X , <B a cover of X, diam d A = 0 i f f A C <r{G G <& : x £ G } f o r each x € A. Proof:, diam H A = 0 d f e i f f sup d*(x,y) = 0 x,yeA 6 I f f d^(x,y) = 0 f o r every x , y £ A i f f i n f dr (x,y) = 0 f o r every x , y £ A GG<6 i f f x,y G G f o r some GG<B f o r every x,yG A i f f A C <r{GG & : xGG} f o r every x G A. Proof of 15.8.1: I f (B G cover Jb or c o v e r - ^ or c o v e r 2 i ^ , then B C G f o r some GG(B Implies diam d B = 0, by lemma 15.11. Proof of 15.8.2: L e t <8 G cover £ and choose an open cover g , a s t a r - r e f i n e m e n t of <£> . Then by lemma 15.11, diamd B = 0 i m p l i e s B i s contained i n the s t a r at x of g, 6 f o r any x £ B . Since !/ Is a s t a r - r e f i n e m e n t of (& , B C G f o r some G € <6 . Hence and s i n c e <fe was an a r b i t r a r y element of coverJt , Proof of 15.10: Suppose ( B G c o v e r J r and H f e 6 l ^ ^ , M f e £ , and x £ X . We know from the pr o o f of 12.3.3 t h a t H^tx] = ( r { G G <& : x £ G } . By d e f i n i t i o n s 6.1.1 and 15.9, and lemma 15.11, M^tx] = (TJACX : x £ A and A C < r { G £ (B : x £ G } } = <r{GG <& : x £ G } = H ^ x ] . 15.12 EXAMPLE. A case In which <PA>rjA. Let G]_, G 2 , G 3 be subsets of X such t h a t X = G 1 U G 2 U G 3 j G 1 - ( G 2 U G 3 ) / 0 , G 2 ~ ( G 1 U G 3 ) / 0 , G 3 ~ ( G - L U G 2 ) / 0 ; G^fl G 2 - G 3 / 0 , G - L P I G 3 ~ G 2 / 0 , G2C\G^^G1 / 0 . Let { G ^ , G 2 , G 3 } be the subbase f o r a topology f o r X. Let r B = 10 i f B = 0 } l i f B / 0 , and A = (G 1flG 2) U ( G 2 r i G 3 ) U ( G ^ G x ) . Now any open cover of X must have G-p G 2, and G^ as elements. Since two of these are necessary and s u f f i c i e n t t o cover A, we have ^ A = 2 f o r each <& £ c o v e r t , and hence f A = 2. On.the other hand, by lemma 15.11, diarn^ A = 0> and so ri^A = 1 f o r each & £ cover & and CHAPTER I I I MEASURES ON QUASI-UNIFORM SPACES In t h i s chapter we c o n s i d e r three methods of g e n e r a t i n g a measure on a q u a s i - u n i f o r m space, and so, s i n c e every t o p o l o g i c a l space i s q u a s i - u n i f o r m i z a b l e ( 3 . 8 ) , on any topo-l o g i c a l space. We show t h a t these measures i n c l u d e f, A, and ^2> and i n c e r t a i n cases CP. When we r e s t r i c t the. qua s i -u n i f o r m i t y t o a u n i f o r m i t y , the three methods r e s u l t i n the same measure jx and we apply some theorems from chapter I t o o b t a i n m e a s u r a b i l i t y p r o p e r t i e s of jx. Theorem 1 8 , 3 . 2 , on m e a s u r a b i l i t y of c l o s e d J}^ s e t s , i s an important r e s u l t f o r the development i n chapter IV. Throughout t h i s chapter we assume g i v e n (X, U.), a q u a s i - u n i f o r m space, and X, a~ gauge on (X, a f a m i l y of subsets of X such that 0 E (X. 1 6 . The measures yu, jx^, and Jjf. We now d e f i n e three types of c o v e r i n g s e t s i n terms of the q u a s i - u n i f o r m i t y , and use them to generate the measures jx, JJJ , and^U. . R e c a l l the d e f i n i t i o n s i n s e c t i o n 3 of chapter 0 . 1 6 . 1 DEFINITIONS. For U G X X X , .1 U* = { A C X : A X A C U | , U + = { A C X : f o r some x £ X, ACD[x']}, xft = { A C X : f o r some VEfy, A X V [ A ] C . u } . For A C X , .2 / I J JA = i n f { ] T < t B : & i s . countable, < 6 C U * ( 1 f l , B€fi and A C <r&}. .3utyA = i n f f ^ r B : 6 . i s countable, <&CU tn&, and A C cr(B}. u +A = sup L U A . ^ UeiT u .4 M # A = i n f { X N B : & i s countable, & C U^f| # , ' u Be© and AC<T<6]. jjft A = supjL^A. 1 6 . 2 REMARK. The same measures are obtained i f the supremum i s taken over a base f o r %l, i . e . i f 1/ i s a base f o r the q u a s i - u n i f ormity ^ , and f o r A C X , 9 A = sup/i rA, vey v then 8 = jut. Proof: Since If a l l , we have Given any U G K , there e x i s t s V E V such t h a t VC'U. Hence V C U and so and s i n c e U was an a r b i t r a r y element of IX, 16.3 REMARK. Pe r v i n [ l l ] p o i n t s out that two non-comparable q u a s i - u n i f o r m i t i e s f o r X may g i v e i d e n t i c a l t o p o l o g i e s f o r X. They may a t the same time y i e l d d i f f e r e n t measures. In the case i n 16.5 below, we note t h a t (3.6) i s j u s t the me t r i c topology. In the case i n theorem 17.1 below, a p p l i e d i n a met r i c space, ^ i s by c o n s t r u c t i o n again the met r i c topology, but we have seen that f and A do not always agree on a metric space (see remark 14:3). ' 16.4 REMARK. The same s i t u a t i o n i s tr u e f o r u n i -f o r m i t i e s . Let ( X , b e the set of a l l o r d i n a l s l e s s than the f i r s t uncountable o r d i n a l , w i t h the d i s c r e t e topology. Let and ^ 2 = { u : U = A U ( X ~ { x : x < a j ) X ( X ~ { x : x< a } ) f o r some a 6 X j . I t i s easy t o check t h a t li^ i s the base f o r a uniform-i t y f o r X. C l e a r l y % 2 s a t i s f i e s a ) , b ) , and d) of 3.11.6. By lemmas 3.2, f o r any u e K 2 , UoU = U, so c) Is s a t i s f i e d a l s o . The topology induced by each of these u n i f o r m i t i e s i s o b v i o u s l y the d i s c r e t e topology. Now f o r B C X , l e t r B = (0 i f B i s countable [ l otherwise . Let be generated u s i n g ^ a n c i u s i n g 2^ 2> Now suppose A C X Is uncountable. Then jx&A = oo and so ^ A = 00 . On the other hand, f o r any U 6 ^ 2 ' /^2A = 1, s i n c e U = A U ( X ~ { x : x < a } ) X ( X ~ { x : x < a } ) f o r some a € X , and A can be covered by { ( X ~ { x : x < a } ) ] U { x * . x < a } . Hence y?A = 1. 16.5 REMARK, JDL i s a d i r e c t g e n e r a l i z a t i o n - of /. Let X be a m e t r i c space w i t h m e t r i c d. I f we set u*r = { (x,y) : d(x,y) 4 r j , and U = [ur : r > 0 J , then H i s c l e a r l y a base f o r a (quasi-) u n i f o r m i t y f o r X. Since A X A C U r I f f d i a m d A < r , we have !f r = jUy f o r r > 0 . Using remark 5-3 we conclude T=jx. + # 17. P r o p e r t i e s o? jx, JJJ , and jx . We c o n s i d e r f i r s t some p r o p e r t i e s of these measures when Vi i s only a q u a s i - u n i f ormity. 17.1 THEOREM. I f U i s Pe r v i n ' s q u a s i - u n i f o r m i t y , (see 3.8) f o r a t o p o l o g i c a l space (X, A), then jx.= A ( s e c t i o n 1 3 ) . 17.2 THEOREM. I f % i s Pe r v i n ' s q u a s i - u n i f o r m i t y f o r a t o p o l o g i c a l space (X, J r ) , then JJ$ = <f2 ( s e c t i o n 1 2 ) . 17.3 THEOREM. I f 1L i s a q u a s i - u n i f o r m i t y f o r a t o p o l o g i c a l space (X, & ) such t h a t Te^ = , then jj~^ f ( s e c t i o n 12). I f Ii i s the maximal q u a s i - u n i f o r m i t y i n d u c i n g if. on X, and (X, Jj) has p r o p e r t y Q (2.2.4), then j j = <P. .17.4 REMARK. We can have / f i n a space having p r o p e r t y Q, If i s obtained u s i n g P e r v i n ' s q u a s i - u n i f o r m i t y f o r the space i n example 12.7,1 (see a l s o 2.3.4), then We now co n s i d e r some p r o p e r t i e s of these measures when *U i s a u n i f o r m i t y . 17.5 REMARK. U n l i k e the case f o r a q u a s i - u n i f o r m i t y , the same measure i s obtained i n the u n i f o r m i t y case whether * 4- A . we use covers from U or from U', i . e . JX=JJJ. The f o l l o w i n g theorem i s analogous t o a s i m i l a r w e l l known r e s u l t f o r f-measure ( s e c t i o n 11) : /A i s the same whether we r e q u i r e of the c o v e r i n g s e t s i n d e f i n i t i o n 11.1 tha t diam Bj_4 6 or diam Bj_< 6 . (See, f o r example, the f i r s t sentence on p. 147 of Sion and Sje r v e [13].) 17.6 THEOREM. I f HJL i s a. u n i f o r m i t y , then jx=jJif. We have seen t h a t a u n i f o r m i t y *U i s c h a r a c t e r i z e d by a f a m i l y of pseudo-metrics, the gage of Ii (3.11.7). The d e f i n i t i o n of \f i n s e c t i o n 11 i s v a l i d f o r a pseudo-m e t r i c . The measures generated u s i n g the pseudo-metrics i n the gage of UL can be used to o b t a i n JJL. 17-7 DEFINITIONS. Let I i be a u n i f o r m i t y f o r X, G the gage of <U • For jD€G, A CX, oc« nA = i n f { X J r B , : A C l J B i » and f o r each iGu>, i€60 ieco B j E & and d i a m B ^ l / n J . oc0 A = sup a. A. ? n€0) ^ n a A = sup ocJ\. jD€G r .17.8 THEOREM. I f G i s the gage of fy , then oc = jx. We now examine the r e l a t i o n s between jx, and r\ and *f of s e c t i o n s 15 and.12. 17.9 THEOREM. I f H i s a u n i f o r m i t y f o r a t o p o l o g i c a l space (X, i i ) such t h a t = A , then jx^r\_. I f f u r t h e r , fy c o n s i s t s of a l l neighborhoods of A , then jx = V\. 17.10 THEOREM. I f (X,& ) i s a paracompact t o p o l o g i c a l space, and H i s the maximal u n i f o r m i t y i n d u c i n g & on X, then f = ^ = JJ,. PROOFS AND EXAMPLES Proof of 17.1: A p p l y i n g theorem 15.4, we show JJL=<?2> the measure d e f i n e d i n 15.3. JX>^2 ' L e t & € cover 2J^., (6 = G = {Gx,...,Gn}, G^E f o r 1 = l , . . . , n . Let n u = O s G i = l i . where S Q = (GXG) U ( ( X - G ) X X ) . (see 3.8) Then U £ K . Now f or G G & , A X A C S Q i f f A C G or Ap|G = 0. Hence A X A C U ( A E U ) i f f A i s contained, i n each G±£ & which i t . i n t e r s e c t s . But then diampQ^A = 0 .for i = l , . . . n , whence diams A = 0, and so ^G, 0 = % ' Since <& was an a r b i t r a r y member of coverg*^, we have ^ 2 ^ ^ : Let U E l t . Then there e x i s t s V G K , V C U such that „ ' V = f l s G j = l J .where G J E J # f o r j=l,...,m. Let S = G = { G ^ E cover 9 ) A . L J Jj=l,...,m 2 L e t 6 < l / 2 m , and diam s B<6. Then f o r J=l, ...,m. G diany) B = 0 and so BCG j_ i f f BDG^ / 0 . Hence m B X B c P l S G , or B £ V . Thus any set of S Q-diameter. l e s s than d i s an •* element of V and we have and s i n c e U was an a r b i t r a r y element of UL , Proof of 17.2 : jjf ^  f 2 : Let (6 E cover 2»#, <6 = {G-p ... , G n ] , where G^E f o r i = l , . . . , n . Let n u = C]sG e U. i = l i Suppose AG U * . Then f o r some x £ X , A C U [ x ] . Now x £ G j f o r some GjG (B , and s i n c e U C S Q ,-we have U[x] C S G [x] = Gj . Hence A C G j . a n d so 6 C U1". i m p l i e s 8 r e f i n e s (6. There f o r e and J^^2 ' Suppose U G ^ . Chose V G ^ i , V C U , m V where G < G & f o r j=l,-...,m. Let <& = f v [ x ] = x G XJ. Now f o r xG X, e i t h e r V[x] = X, or k V[x] = P | Q i f o r some k, l < k < m , i = l J l and.some f u n c t i o n j on {1,...,m] onto {l,...,m| , f o r : Suppose x^Gj_ f o r i=l,...,m. Tnen m V[x] = ( f W ) [ X ] i = l U 1 = {y: ( x , y ) G ( G i X G i ) U (.(X~G ±) X X) f o r 1=1,. . .,m } = X. On the other hand, suppose xGGj_^ f or i = l , . . . ,k, l ^ k ^ m , and x^G - s f o r i = k+l,...m, f o r some f u n c t i o n 1 j on {1,...,m| onto {l,...,m|. Then ( x , y ) G V i f f (x,y) G G 1 X G 1 f o r 1=1,... ,k J i J l i f f y G G 1 f o r i = l , . . . , k J i k i f f yePk^ > i = i H v[x] = n G . I • 1=1 J i We conclude t h a t each element of & i s open and & i s f i n i t e . Clearly<fe i s a cover of X, so 6 G c o v e r 2 ^ . T r i v i a l l y , i f A C B f o r some B £ ( 6 , then A C V [ x ] f o r some x EX, so 6> r e f i n e s (6 i m p l i e s S C and hence and Proof of 17.3: Suppose fy i s a q u a s i - u n i f o r m i t y f o r (X, A ) and ^ = ^ . Let UG fy . For each x G X, U[x] i s a neighborhood of x so there e x i s t s open G X such t h a t x G G X C U[x ] . L e t (B = { G X : x G X } . Then & G c o v e r t and i f £ r e f i n e s <fe, S C U ^ , so \ >A and Suppose now fy i s the maximal q u a s i - u n i f o r m i t y i n d u c i n g Jf on X, and. (X, J#) has p r o p e r t y Q. Let &G c o v e r t and l e t & G c o v e r t , $ r e f i n i n g £ and such t h a t T f { G G & xGG} i s open f o r every x G X . Set U = P|s G G€6 where agai n S G = ( G X G ) U ( ( X ~ G ) X X ) . Then 1 ) f o r every x GX, U[x] i s a neighborhood of x, and 2) UoU = U. 1 ) : We show U[x] = T T {G G <6 : X G G } . y G U[x] i f f ( x , y ) G U i f f ( x , y ) G S G f o r every GG<6 I f f ( x , y ) G G X G or ( x , y ) € ( X - G ) X X f o r every GG(B i f f x , yGG or x ^ G f o r every G G f i i f f y E G , f o r each G G f i such t h a t x G G i f f y G Tr{G & f i : x g G J . 2) : By d e f i n i t i o n , UoU = {(x,y) : f o r some z , ( x , z ) G U and ( z , y ) £ I l } , Now i f ( x , y ) G U , then s i n c e (x,x)G U, we have (x,y)G UoU, and so UCUoU. Suppose now (x , y ) G U o u . Then f o r some z, ( x , z ) G U and ( z , y ) G U . Hence ( x , z ) G S Q and ( z , y ) G S G f o r every G E & . Let G G f i . a) i f ( x , z ) G G X G and ( z , y ) G G X G , then (x,y)G G X G C S Q . b) i f (x,z) G ( X ~ G ) X X and (z,y) G ( X ~ G ) X X , then ( x , y ) G ( X ~ G ) X X C S G . c) I f (x,z) e ( X ~ G ) X X, (z,y) G G XG, then (x,y) G ( X - G ) X x c s G . d) ( x , z ) G G X G and (z,y) G (X~G ) X X i s i m p o s s i b l e . Hence ( x , y ) G S G f o r every G G <B and so ( x , y ) G U and we have UoUCU. Now 1 ) i m p l i e s t h a t A C U , and t h i s w i t h 2 ) i m p l i e s t h a t {u} i s the base f o r a q u a s i - u n i f o r m i t y f o r X. Hence by theorem 6 . 3 of K e l l e y , ZCU{u} i s the subbase f o r a q u a s i -u n i f ormity V f o r X. But ^ = >it (the proof, ; u s i n g l ) , i s e s s e n t i a l l y the same as t h a t i n 3 . 1 0 ) , and so s i n c e Vi i s the maximal q u a s i - u n i f ormity i n d u c i n g J/ on X, we have V = U and U G ^ . Now i f A C U [ x ] f o r some x G X ( i . e . A G ), then by the pro o f of l ) , A C G f o r each G G <B such t h a t x GG, and hence ^ C U * i m p l i e s X. i s a refinement of (& . There f o r e A and we have Proof of 1 7 . 5 : ^jX : Suppose UG'W, and l e t A X A C U , and x,yGA. Then (x,y)GU,and so y G U [ x ] , Hence A C U [ x ] f o r every x G A and U C U 1 , and we have y ^ u 1 and jJL^jjf: Given U G H , u s i n g 3 . 1 1 . 4 and 3 . 1 1 . 6 , c ) , choose symmetric VG*U such t h a t VoVoVCU. A p p l y i n g 3 . 1 1 . 5 we t * conclude V ' C U . Hence and Proof of• 17.6: y.<:jjf : Given U G K , l / c u * , S O : Let UGH. Chose V E ^ . such t h a t V i s sym-me t r i c and VoVoVCU. Suppose A E V . Then A X A C V and by 3.11.5, V [ A ] X V [ A ] C U . Hence A E and we have and Proof of 17.8: For pEG, l e t vj>,n = {( x>y) : p(*>y) < 1/n-f. Then by theorem 6.19 of K e l l e y ^ = { vp,n : i>^G, n e w ] Is a base f o r % . By remark 16.2 jx may be d e f i n e d u s i n g V. Now f o r A C X , L N F { Z I ^ B i : A C l j B i . a n d f o r i E c o , ) G G X W iew iew ocA = sup sup oc 0 nA = sup oc A peG new J' (p,n)eGxco P , n = sup (J3,n e BjEcJl and d i a m p B i ^ 1/n j = sup i n f { J ] r B i : A C W » B i a n d f o r i£u), (p,n)GGxw iew B ± ea and B i X B j C V 5 s u p r ^ A ,^A. Proof of 17.9: Suppose Ii i s a u n i f o r m i t y i n d u c i n g ^ on X. Let U € l i . For each x £ X , u s i n g remark 3.11.3 choose an open neighborhood G x of x such t h a t G X X G X C U , and l e t & = {G x : x E X J G c o v e r t . Then I f diama A = 0, then A X A C U , f o r suppose diarn^ A = 0, © © ( x , y ) E A X A =^d 6(x:,y) = 0 => f o r some GG ( 6 » ^ G(x»y) = 0 => f o r some GG<B> x , y £ G => f o r some G G<6 , (x,y) E G X G = > ( x , y ) G U . Hence and Now suppose f u r t h e r t h a t Ii c o n s i s t s of a l l neighbor-hoods of A . L e t B G c o v e r t . Let U = <r{GXG : G E (6} . Then U G l £ . Sippose A X A C U . Suppose x £ A and l e t y E A . Then ( x , y ) E A X A and so ( x , v y ) G G X G f o r some G G f i . Hence y G <r{G G B : X G G } , and we have ACcr{GG<& ' x G G J f o r each x G A . By lemma 15.11, diam d A = 0. We conclude t h a t and Proof of 17.10: A paracompact space i s completely regu-l a r and the maximal u n i f o r m i t y c o n s i s t s of a l l neighborhoods of A (3.11.9). By theorem 17.9, JJL= ^ . A r e g u l a r space i s paracompact i f f i t i s f u l l y normal (2.3.2). By theorem. 15.8.2 then, = (We c o u l d r e p l a c e 'paracompact' by ' r e g u l a r and f u l l y normal' in. the hypothesis of t h i s theorem.) I7.II EXAMPLE. A case where <P = Y\ £ JX. In the example of 16.4, U± i s c l e a r l y a base f o r the maximum.uni-f o r m i t y f o r X. A l s o the space Is metric,. and hence para-compact, so we have f = ^ = jx1 = ^f. However, we saw there t h a t y U 2 / JJ?-, showing, t h a t the h y p o t h e s i s t h a t Vt consist, of a l l neighborhoods of A i s r e q u i r e d f o r theorem,17.10. In t h i s l i g h t we note a l s o t h a t d i f f e r e n t , u n i f o r m i t i e s f o r a m e t r i c space may r e s u l t i n d i f f e r e n t measures. 18. M e a s u r a b l l i t y theorems. In t h i s s e c t i o n we r e s t r i c t our a t t e n t i o n t o the u n i f o r m i t y case and obtainsome m e a s u r a b l l i t y theorems f o r j x . To apply the t h e o r y of chapter I,we l e t 18.1 DEFINITIONS. Hu = { A C X : A X A C U } , % = { % : U € 7 4 Then 04 i s a f i l t e r b a s e i n X and JX='V^,z\ P r o p e r t i e s of 04 and the ^ - t o p o l o g y are i n d i c a t e d i n the f o l l o w i n g theorems. 1 8 . 2 THEOREMS. . 1 04 s a t i s f i e s ( 7 1 1 ) . . 2 I f fy c o n s i s t s of a l l neighborhoods of A , then 04 s a t i s f i e s ( 7 I I I ) . . 3 The 04-topology i s the u n i f o r m topology, 3*^, • A p p l y i n g the r e s u l t s of chapter I we then get the f o l l o w i n g m e a s u r a b i l i t y theorems. 1 8 . 3 THEOREMS. . 1 Compact j^fj s e t s are yu-measurable. ' . 2 I f the u n i f o r m topology i s compact, then c l o s e d ^ s e t s are^x-measurable. . 3 I f IX c o n s i s t s of a l l the neighborhoods of A , then c l o s e d Jj^ s e t s are JJ, -measurable. . 4 I f the u n i f o r m topology i s paracompact, and fy i s the maximal u n i f o r m i t y , then c l o s e d jj^ s e t s are yUL-measurable. PROOFS AND EXAMPLES 1 8 . 4 LEMMA. I f U 6 ^ , U I s symmetric, and x £ X , then Hr;[x] = U [ x ] . Proof: Using d e f i n i t i o n s 6 . 1 . 1 and 3 . 4 . 2 , y E H y f x ] i f f ye A f o r some A £ Hg. such t h a t x £ A i f f y £ A f o r some A such t h a t A X A C U a n d x £ A i f f {x,y} X U > y j C U i f f (x,y)£ U i f f y £ U & x ] . Proof of 18.2.1: Suppose . Then H = Hy f o r some U E U. Choose symmetric V, W £ K such that V C U and WoWCV. Suppose A C X . Then by lemmas 6.1.2, 18.4, and 3.5.1, HV[.A] = U H V [ X ] = I J v [ X ] = V [ A ] . X€A X€A S i m i l a r l y % [ A ] = W[A] and H^A] = U[A], But by 3.5.3, W[W[A]] = (WoW)[A]CV[A] . Hence H W [ H W [ A ] ] C H V - [ A ] C H U [ A ] . Proof of 18.2.2: Suppose A C X i s c l o s e d , B C X i s open and A C B . Let U = B X B U ( X ~ A ) X ( X - A ) . Then U, i s a neighborhood of A and so XJEU. I t i s c l e a r t h a t U[A] = B. Proof of 18.2.3: D e f i n i t i o n s 6.1.3 and remark 3.6 show t h a t these two t o p o l o g i e s are d e f i n e d In the same way, one u s i n g Hrj[x] and the other U [ x ] . The d e s i r e d c o n c l u s i o n f o l l o w s from remark 3.H.4 and 18.4. Proof of 18.3.1: Use 18.2 and theorem 9.6. Proof of 18.3.2: The c o n c l u s i o n i s an,immediate c o r o l l a r y of 18.3.1. Proof of 18.3.3: Use 18.2 and theorem 9.7-Proof of 18.3.4: I t has a l r e a d y been observed In the proo f of 17.10 t h a t i n t h i s case the maximal u n i f o r m i t y c o n s i s t s of a l l neighborhoods of A . The c o n c l u s i o n i s then a c o l o l l a r y of 18.3.3. 18.5 EXAMPLE. A non-measurable c l o s e d s e t . The space of 16.4 i s me t r i c and so paracompact. I t . i s not compact. i s a base f o r a u n i f o r m i t y which does not c o n s i s t of a l l neighborhoods of A . i s a base f o r the u n i f o r m i t y con-s i s t i n g of a l l neighborhoods of A . We saw. t h a t i f AGX.was uncountable, then y?-k = 1. I f the complement of A i s a l s o uncountable, then A i s not ^i 2-measurable. But any.subset of X . i s a c l o s e d ^ s e t . Any compact subset, of X i s f i n i t e and so h a s ^ A . 2 _ m e a s u r e zero and i s ju 2-measurable. JJL?- i s a 0-00 measure and. a l l subsets of X are ju,1-mea sur a b l e . CHAPTER IV. MEASURES ON A COMPACT HAUSDORFF SPACE The purpose of t h i s chapter i s t o i n v e s t i g a t e the p o s s i b i l i t y of extending c e r t a i n r e s u l t s obtained i n a com-pact m e t r i c space by Slon and Sjerve [ 1 3 ] ( h e r e a f t e r r e f e r r e d t o simply as Sion and S j e r v e ) . We work i n a compact Hausdorff space and generate the measure j j i ( l 6 . 2 ) u s i n g . t h e u n i f o r m i t y f o r the space and a gauge % r e s t r i c t e d as i n Sion and S j e r v e . We could assume a compact r e g u l a r space but the Hausdorff assumption s i m p l i f i e s the develop-ment, and can be made w i t h , l i t t l e l o s s of g e n e r a l i t y (see remark 2 0 . 3 ) . A f t e r I n t r o d u c i n g the r e s t r i c t i o n on x, we o b t a i n some p r o p e r t i e s of jx, a major one being r e g u l a r i t y (theorem 2 0 . 6 ) . In s e c t i o n 2 1 the p a r t i a l o r d e r i n g >- on gauges i s i n t r o d u c e d . A good d e a l of work i n Hausdorff h-measure theory has been a s s o c i a t e d . w i t h t h i s concept and non-<r-f i n i t e n e s s . Theorem 2 1 . 3 shows t h a t i t s u s e f u l n e s s i s e s s e n t i a l l y r e s t r i c t e d t o the m e t r i c case. We conclude w i t h a theorem and example which begin, an i n v e s t i g a t i o n of a theorem on non-<T-finiteness which does not i n v o l v e the o r d e r i n g >-. 1 9 . P r e l i m i n a r i e s . Given the u n i f o r m i t y OJL f o r X, we now . i n t r o d u c e the topology induced by 2l on J , the f a m i l y of subsets of X, and o b t a i n some simple f a c t s about i t . 1 9 . 1 DEFINITIONS. For A G J and U e ^ , . 1 Ny(A) = { B C X . : A C U [ B ] and B C U [ A ] ( , . 2 VA = ( w C J : f o r some UG *2Z , WDN^A)}, . 3 (k = {W : W G Vk f o r each A £ W}. 1 9 . 2 THEOREM, ft i s a topology f o r J , and f o r AG J , 1/h i s the neighborhood system of A r e l a t i v e t o & . 1 9 . 3 REMARK. Michael [8] induces a topology on the subsets of a un i f o r m space by the same process as that, above except t h a t he s e t s Nu(A) = J B C X : B C U [ A ] and B D U [ x ] ^0 f o r a l l x £ A J 1 9 . 4 LEMMA. Michael's topology on J i s Just & . 1 9 . 5 REMARK. In the non-compact s i t u a t i o n , i t i s p o s s i b l e t o have two non-comparable u n i f o r m i t i e s which induce the same topology on X, but non-comparable t o p o l o g i e s on J . In the compact case the u n i f o r m i t y i s unique and the topology on J i s determined by t h a t on X. 1 9 . 6 LEMMA. I f X i s compact i n 3 ^ , then the f a m i l y y of s e t s c l o s e d In ^ i s a compact subset of J i n We now i n t r o d u c e some r e s u l t s on a p a r t i c u l a r k i nd of convergence i n J . 19.7 DEFINITION. Suppose ( B « , «-G D) I S a net In J . Then B =<fl-lim B a i f f B G ^ and f o r every UEV., there oc e x i s t s j8GD such t h a t f o r oc»j3, B^G Ny(B). (see 1.1-7) . 19.8 THEOREM. I f X i s compact,in "7^, then f o r every net ( B a , ocG D) i n J , there i s a subnet (Bp, ;6GE) and B e ? such t h a t B =<ft.-lim B A . f> P 19.9 LEMMA. Suppose (A*, otGD). i s a net i n J , A = i f t - l i m A K, U G ^ , U i s c l o s e d i n the product topology, and A^XAx-CU f o r every <x£D. Then A X A C U . PROOFS Proof of 19.2: The c o n c l u s i o n f o l l o w s from p r o p o s i t i o n 2, chapter I of Bourbaki [3] a f t e r we have checked the f o l l o w i n g : 1) I f W G VA, then AG W, s i n c e A G N ^ A ) f o r every UG IK . 2) I f W G VA> a n d YDW, then Y G Y A by d e f i n i t i o n of VA. 3) I f W,YG74, t h e n W f l Y G ^ A . Suppose W, Y G ^ . Then there e x i s t U G ^ such that WDU[A] and V G H such t h a t Y D V [ A ] . Hence WflY D U [ A ] f | V [ A ] D (unv)[A] by lemma 3.5.2. B u t UDVG.W, s o W f l Y G ^ . 4) I f W G then there e x i s t s Y G V A. such t h a t YCW and f o r each BGY, WG*ZA>. 94 Since W E V A , there e x i s t s U E U such that WDNu(A). Choose symmetric VEU, VoVCU, and set Y = Ny(A). a) . Y C W : f o r l e t B E N V ( A ) . Then B C V [ A ] and A C V [ B ] . But V C U , so B C V [ A ] C U[A], A C V [ B ] C U [ B ] . Hence B e % ( A ) C W . b) For B E Y + Ny(A), we have W 6 VB: Suppose B E N v ( A ) . Then B C V [ A ] , A G V [ B ] , Let C E N B ( B ) . Then C C V [ B ] and B C V [ C ] . Hence C C V [ B ] C V[V[A]] = ( V o V ) [ A ] C U[A], and A C V[B] C V[V[C]] = (VoV)[C] CU[.C.]. Hence C E N u ( A ) and so N V(B) C % ( A ) C W . Theref o r e WE'Z/g. Proof of 19.4: I f U; i s symmetric, -then A C U [ B ] i f f f o r a l l x E A there e x i s t s y E B such t h a t ( y , x ) E U ; i f f f o r a l l x E A th e r e e x i s t s y E B such t h a t ( x , y ) £ U i f f f o r a l l x E A there e x i s t s y E B such t h a t y E U[x] i f f f o r a l l x £ A, B f l U [ x ] / 0 , and the two d e f i n i t i o n s of N^A) are e q u i v a l e n t . But the symmetric members of I i form a base f o r ^ (3.11.4) and hence u s i n g only symmetric members,,we get the same base f o r the neighborhood system of A E J i n each case, and the t o p o l o g i e s are the same. Example for-19.5: Let X = (0,Co ) w i t h the u s u a l topology, U be the u n i f o r m i t y having as a base a l l elements of the form {(x,y) : |x-y| < r} f o r some r > 0 , V the u n i f o r m i t y having as a base a l l elements of the form f( x , y ) : |x/y - l | < r } f o r some r > 0 . Then = 7^ i s the u s u a l topology. We show, however, t h a t there i s a neighborhood <B of X£j i n f l y (the topology induced on J by I f ) such t h a t every neighborhood of X i n f i ^ (the topology induced by I t ) c o n t a i n s p o i n t s i n J not i n <& , and s i m i l a r l y f o r a neighborhood of X i n • Let <& = Ny ( X ) , where V = {(x,y) : |x/y - l | < r ] f o r some r > 0 . Now V [ X ] = X f o r any V € V , so N V ( X ) = { B C X : X C V [ B ] J . Suppose (P i s a neighborhood of X i n &<^. Then f o r some U.in the above base f o r 11, < £ D N u ( X ) = ( B C X : X C U [ B ] J . But U = -j(x,y) : |x-y|<6J f o r some A >0. Choose A = (6,00). Then U[A] = X and A6 Ny ( X ), but V[A] = , » ) X . Hence A G % ( X ) , A ^ N y ( X ) . The c o n s t r u c t i o n f o r the other case, g i v e n <& = Ny ( X ) , U = {(x,y) : |x - y| < r} f o r some r > 0, and any neighborhood of X i n fa^ i n v o l v e s p i c k i n g a set A whose p o i n t s get a r b i t r a r i l y f a r apart as t h e i r numerical v a l u e s i n c r e a s e , w h i l e A s t i l l i s an element of P . I t cannot, o f c o u r s e , be an element o f (fe . Pr o o f o f 19.6: Theorems 3.3 and 4.2 o f M i c h a e l [8] prove t h e lemma f o r h i s t o p o l o g y on J . The c o n c l u s i o n f o l l o w s by u s i n g the f a c t t h a t our t o p o l o g y a g r e e s w i t h h i s (19.4). P r o o f of 19.8: By 19.6 t h e theorem,holds f o r any.net i n T . L e t (B^, O C E D) , be a net. i n J and c o n s i d e r the net ( B a , ocED). L e t ( B ^ , j B E E ) \ b e a subnet and B G T Such t h a t B = & - l i m Bo. Now suppose U E 1 / . Choose V E ^ t , V o V C U . Then t h e r e e x i s t s y E E such t h a t f o r a l l p » y . . B C V [ B ^ ] and B ^ C V [ B ] . Then B^ C U[B] and using.3.11,12, BCV[1\>] =V[Ou[Ba]] C V[V[B«]] F Veil • • • = (VoV)[Bp] C U [ B f ] . Hence B = k - l i m Bo. p P P r o o f o f 19.9: Suppose (x,y) ^ U. By 3.11.13 U =f~N|VoUoV, V<Z11 so f o r some symmetric V£ll, ( x , y ) ^ V0U0V. Now X V C U f o r e v e r y ocED. Hence by 3.11.5 V [ A K ] X V f A ^ ] C Vouov f o r e v e r y ocED. T h e r e f o r e f o r e v e r y ocE D, (x,y ) $ V f A * ] X V f A * ] , and .so-by . d e f i n i t i o n . 19.7, (x,y) $ A X A . 97 20. The f a m i l y T. In t h i s s e c t i o n ( X , i ) i s a compact H a u s d o r f f space, %L i s t h e u n i f o r m i t y f o r X w h i c h i n d u c e s A , J Is the f a m i l y of a l l s u b s e t s of X, % i s a gauge on J , and JJL i s the measure of c h a p t e r I I I g e n e r a t e d by % and z . We now i n t r o d u c e the f a m i l y o f s e t f u n c t i o n s t o w h i c h we w i l l r e s t r i c t X, and examine some sequences o f t h i s r e s t r i c t i o n . 20.1 DEFINITION. T i s the s e t o f . a l l f u n c t i o n s X on J such t h a t .1 t'0 = 0; .2 i f A G B , then t A < t B 4 00 j .3 t'A = 0 A X A C A ; .4 T1 i s c o n t i n u o u s i n J w i t h r e s p e c t t o t h e t o p o l o g y k ( 1 9 . 1 . 3 ) ; and .5 T* i s bounded on . 20.2 REMARKS. .1 I f ( X , i ) i s a compact m e t r i c space t h e n our f a m i l y T and measure jx a r e j u s t t h e f a m i l y T and measure of S i o n and S j e r v e . .2 The f a c t t h a t t h e topology, i s compact g u a r a n t e e s b o t h t h a t the f a m i l y T i s de t e r m i n e d by the t o p o l o g y on X (see 19.5) a n d . t h a t g i v e n t E T , the measure JJU i s d e t e r m i n e d by the t o p o l o g y on X, s i n c e t h e r e i s a unique u n i f o r m i t y . 2 0 . 3 REMARK. Suppose (X, Jr) i s a compact, r e g u l a r , but not Hausdorff, and t G T . Then a consequence of c o r o l l a r y 2 0 . 1 0 i s th a t f o r A CX, MA = u(AUUci{x}). y xeA Thus the measure does not d i s t i n g u i s h between p o i n t s and t h e i r c l o s u r e s , and so without any r e a l l o s s of g e n e r a l i t y we may I d e n t i f y each p o i n t w i t h i t s c l o s u r e , g i v i n g a T-^ , and hence Hausdorff space s i n c e the space i s r e g u l a r . For t h i s reason we assume from the s t a r t t h a t the space i s Hausdorff. The next two theorems extend s i m i l a r theorems f o r Hausdorff h-measures. 2 0 . 4 THEOREM. I f t € T , then the r e s t r i c t i o n of x t o the f a m i l y of open s e t s generates the same measure J J L as x . 2 0 . 5 THEOREM. I f T E T,•then the r e s t r i c t i o n of x t o the f a m i l y of c l o s e d Jj^ s e t s generates the same measure jx as X. . 2 0 . 6 THEOREM. I f TET and X = LJ An> w h e r e f o r e a c h new nEo), A n E and j j J \ n < oo , then JL*. I S a r e g u l a r measure. The p r o p e r t y of the pre-measure jx^ g i v e n i n the next theorem Is a p a r t i a l e x t e n s i o n of theorem 5 . 2 In Sion and Sj e r v e . A d i r e c t e x t e n s i o n would c o n t a i n no r e f e r e n c e t o V, but V entered as a consequence of the f a c t t h at c l o s e d s e t s are not n e c e s s a r i l y c l o s e d Jj^ s e t s , and we were not 99 a b l e t o e l i m i n a t e i t . 20.7 THEOREM. I f TGT; X = L j B n , .where f o r each nGw, new B n E %p and jxBY)< 00 ; A. i s an ascending sequence of subsets of X; and U,VG U, then ^VQUovL/n 4 l l t n M « ' new n PROOFS Proof of 20.2.1: For the e q u a l i t y of the f a m i l i e s T, we need only check that our topology & i s the same as the subset topology i n Sion and S j e r v e . T h i s f o l l o w s from the f a c t t h a t the u n i f o r m i t y i s unique and th a t there i s a base c o n s i s t i n g of elements of the form {(x,y) : |x-y| < r ] f o r s o m e r >0. For the equality, of the measures we note t h a t our measure ~f i s j u s t the measure y[x) of Sion and Sj e r v e , and by remark 16.5, T= jx • 20.8 LEMMA. I f T G T , ACX, and £ > 0, then there e x i s t s U G 16. such that rU[A] ^  xA + e . Proof: I f XA = 00 , the c o n c l u s i o n i s t r i v i a l . Suppose XA <oo . By 20.1.4 there i s a neighborhood V of A i n J such t h a t f o r B E V, | XA - x B | < £ . By d e f i n i t i o n 19.1.2 and theorem..19.2, there e x i s t s U G H such t h a t V D N y ( A ) . But U[A] G % ( A ) , whence, s i n c e % i s i n c r e a s i n g ( 2 0 . 1 . 2 ) , r U [ A ] - r A ^ 6 . 2 0 . 9 COROLLARY. I f T E T, A G X, . then there e x i s t s a sequence U i n iL such t h a t %k = t f l U n [ A ] , new Proof: I f rA = oo , the c o n c l u s i o n f o l l o w s from the f a c t t h a t x i s i n c r e a s i n g ( 2 0 . 1 . 2 ) . Suppose rA <00. Using lemma 2 0 . 8 , choose a sequence U;in fy, such t h a t •5U n[A] 4 XA.+ 1/n f o r each n ^ 1. Then u s i n g the f a c t t h a t x Is i n c r e a s i n g , we have Zk = r f | u n [ A ] . new 2 0 . 1 0 COROLLARY. I f t £ T , ACX, then r A = xT. Proof: Using 2 0 . 9 choose a sequence U.sueh t h a t TA.= r n u n [ A ] , new But A C A = f"V[A] C D^iAh UGU new Again s i n c e r i s i n c r e a s i n g we have TA = TA~. Proof of 2 0 . 4 : We use the f a c t t h a t jjf (theorem ,17.6) and d e f i n i t i o n 1 6 . 1 . 4 . Let V be a base of symmetric members of fy. Let U E V, A C X , and suppose jj^j A < 00 ,. the argument being t r i v i a l otherwise. Let £ > 0 . Choose B-^EU^ f o r I E w such t h a t A C l j B i 1G6J 101 and X > B ± ^ Ljlk + 6 / 2 . lew u Now f o r each I E CO, by d e f i n i t i o n 1 6 . 1 . 1 and symmetry of U, there e x i s t s Uj_E 1/ such t h a t U±[B±] X V±[B±) C U. Using lemma 2 0 . 8 choose f o r each i E w , V±G V such t h a t v i ° v i C I J i a n d ZV±[B±] ^ + 6 / 2 i + 2 . Then f o r each i E t o , choose Gj_E such t h a t B i C Q i C V i d i ] . Then and. s i n c e x i s i n c r e a s i n g , XG± < xB± + £ / 2 i + 2 f o r each. lew. Hence # X > G i < X > B i + ^ 2 ^ > H T A + £ -ieco ieco A l s o f o r each i G w , V±[G±] X V±[G±] C V i t V i t B i ] ] X V i t V i t B i ] ] C U 1 [ B 1 ] X V±[B±] C U, and so G-jE U^f) & f o r each. i G o ) . We conclude t h a t / ^ . i s u n a f f e c t e d by r e s t r i c t i n g x to A , and hence the same i s t r u e f o r yu#. Proof of 2 0 . 5 : - We show f i r s t t h a t f o r ACX, U,a sequence i n 1L, there e x i s t s a c l o s e d Jr^ set E such t h a t A C E c f ^ j U n [ A] . To see t h i s we c o n s t r u c t a sequence V i n "li by r e c u r s i o n such t h a t f o r n£aj, V n i s symmetric, V n C U n , and Y ^ n ^ V r S e t E = f l v n [ A ] . new Then 1) A C E c O u n [ A ] i s obvious. new 2) E i s c l o s e d . I f x(£E, then f o r some n£<d, x ^ V n [ A ] , so X $ V n + 1 [ V n H [ A ] ] , By lemmas 18.4 and 6.3.3, V n + i [ x ] f l V n + i [ A ] = 0 . But V n + ] _ [ x ] i s a neighborhood of x and E C V n + 1 [ A ] so x i s c o n t a i n e d . i n an open set not i n t e r s e c t i n g E. 3) E G ^ . For each nGo) choose G nG >& such t h a t E C V n + 1 [ A ] C G n C V n + 1 [ V n + 1 [ A ] . ] C V j A ] . Then E C P l G n c P l V ^ = E ' new new E = H G n -new # Now suppose U G ^ , U, symmetric, and BG U . Then f o r some V G H , V[B] X V[B] C.U. Choose symmetric W G H , WoWCV. By c o r o l l a r y 20.9 l e t i ^ n l new b e a s e Q u e n c e l n H. such t h a t U^CW and x B = t H V B ] -new Using the r e s u l t above, c o n s t r u c t a c l o s e d Jj^ set E such log t h a t B C E c P | U n [ B ] . neu> Then XB = t E and E6U^ s i n c e ECW[B] i m p l i e s W[E]CW[W[B]] = (WoW)[B] C V[B], and so W[E] X W[E] C V[B] X V[B] C U. Hence f o r any set B g / , there i s a c l o s e d Jjh set E G such that r B = t E , and so jx& i s u n a f f e c t e d by the r e s -t r i c t i o n of x t o c l o s e d Jt^ s e t s . Since jx^ may be d e f i n e d u s i n g a base of symmetric elements of (U, the same i s then t r u e of jjf =jx (17.6). Proof of 20.6: By theorem 20.5, jx can be obtained by r e s t r i c t i n g x to c l o s e d Jj^ s e t s . But by-18.3.2, c l o s e d j£l s e t s are JJ.-measurable, and the c o n c l u s i o n f o l l o w s from c o r o l l a r y 10.5. 20.11 LEMMA. I f -CGT, (A^ocGD) i s a net i n J , and lJLimrA^ = 0, then given UG'U, there e x i s t s jBG D such that f o r oc»J&, A K X A*C U. Proof: Suppose there e x i s t s 11 EH such t h a t f o r every j5G D, there e x i s t s oc»j3 such t h a t A^ X A^CjlU. Then {A* : A^XAocCjlu} i s a subnet of (A^ocGD), say ( A y , T G C ) . By theorem..19.8 there e x i s t s B E T and a subnet ( A p , j 3 E E ) of (Ay, T E C ) and so a l s o of (A^, ocG D) such t h a t B =. & - l i m Ao . io 4 By the c o n t i n u i t y of z, ZB = l i m rA- = 0 . oc Hence by d e f i n i t i o n 2 0 . 1 . 3 , B X B C A . Now choose symmetric V G K such that VoVCU. Then by d e f i n -i t i o n . 1 9 . 7 , there e x i s t s p'£ E such that f o r py>p , A p C V [ B ] . Using 3 . H . 5 and lemma 3.2.2 we have A ^ X A p C V[B] X V [ B ] C V o A P V = VoVCU. But f o r every j3€.E, A ^ X A ^ C t U . T h i s c o n t r a d i c t i o n e s t a b l i s h e s the lemma. 20.12 LEMMA. Suppose the se t s E , P( i,oc)C X, and Vj^ f o r iE(d and ocED, a d i r e c t e d s e t , s a t i s f y the f o l l o w i n g c o n d i t i o n s : i ) There e x i s t s £ E D such that f o r o c » ^ , E C U P U ' * ) ; i i ) xP(i,ot) ^ T P ( i + l , oc ); and i i i ) V ± = & - l i m P(i,oc). oc I f t £ T and X i s jut-measurably ( T - f i n i t e , then given YE 11, f o r each i € w there e x i s t s Bj_ such t h a t V i C B i C Y t V i ] . XB± = XV i, and Z r V i .+ u ( E - U B i ) lim £ rP(i,«.j. lew lew oc lew Proof s L e t a = l i E £<tP(i,oc), 0 0 lew and suppose a<O0. For each i£to, ngw, choose open Wi n such t h a t 1 0 5 V i C W ^ r C Y t V i ] , and T W i , n ^ r V i + l / n + 1 , and such that f o r each n £ w , w i , n + l C W i > n , as Is done i n the proof of theorem 2 0 . 5 except f o r the descend-i n g requirement, which can be met by t a k i n g i n t e r s e c t i o n s . Let B i » f > i , n • new Given £ > 0 , choose a net (oCj^ j E E). i n D such t h a t f o r every j , k € E , o c k » a j i f k » j , and f o r every j £ E , i v ) J ] r P ( i , a 1 ) < a + £ . i€oo J . Then f o r each I E to, . „ . . ( P ( i , 0 C j ) , j E E ) = ( P ( i , j B ) , jBEC) i s a subnet of ( P ( i , o c ) , cCE D ) so by the c o n t i n u i t y of X ZV± = l i m ?P(i,oc) = l i m T P ( I , 6 ) f o r iGW. CCG D p&C Hence and s i n c e £ was a r b i t r a r y , iGw Let b = a - Z r V i . lew For each i£a>, choose n^E w and W.j_ = W ± > n ^ such that r W i < + £ / 2 i Then X! t V i ^ a i m p l i e s l i m rV^ = 0 , whence lew i / a+e . 2 n l i m zW. = 0. i Now l e t U G ^ . Then a) There e x i s t s K-jG co such t h a t f o r i > K x, Wj€ U*. T h i s f o l l o w s from l i m TW, = 0 and lemma 20.11. i 1 b) There e x i s t s KgGto such t h a t f o r a l l y6G C and i ^ K g , P ( i , j B ) £ U . Suppose not, then gi v e n nGw there e x i s t i n > 2 n and ^ G C such t h a t P ( i m / n ) £ u * -But f o r each nGw, s i n c e I] r P ( i , jB n) < a + e, i n > 2 n and the P(i,yS^) are ordered by d e c r e a s i n g X v a l u e s . Hence l i m r P ( i n , B^) = 0, which by lemma 20.11, n c o n t r a d i c t s the f a c t t h a t f o r each nGw, P( i n , / 3 n ) $ U*. c) . There e x i s t s K3G10 such t h a t Let K = max { K x, Kg, K3 [. Now f o r ieco , there e x i s t s J 3 J G C such t h a t f o r j 3 » j S 1 For by theorem 6.33 of K e l l e y , s i n c e V± i s c l o s e d and hence compact, W 1 D V i i m p l i e s there e x i s t s MEH such t h a t 1 0 7 MtV-jJCWi; and V t = k - 1 1 m P(i,B) i m p l i e s by d e f i n i t i o n 1 9 . 7 t h a t there e x i s t s ^ E C such t h a t f o r p»plf P ( l , y 3 ) C M [ V i ] . Choose V E C such t h a t f ° r 1 = 1 , . . .K, and such t h a t K T.\xv± - T P ( I , Y ) | < e, i=o u s i n g V, = k - l i m P ( i , 6 ) and the c o n t i n u i t y of x. Now K E - U w i C UP(i>Y) i = l i>K and f o r i > K, P ( i , r ) E U * by b) . Hence K u ( E - U W i ) ( E - U W i ) E xp(±,r) = ieo> 1=1 i > K K K = 2>p(i,r) - T,xv± + E«Vi - E ^ p ( i , y ) + £ rv ± ietJ lew 1=6 i=o i > K < a + e .-• £ r v i •+ £ + £ ^ b + 3 £ . lew Since U was a r b i t r a r y , we have u ( E - Uw,) < b + 3 £ . y lew Now f o r each i E t o , the r e s t r i c t i o n s on W± used i n o b t a i n i n g the e x p r e s s i o n above are s a t i s f i e d by Wj_ n f o r n ^ n ^ . L e t t i n g N = max [njL : i ^  K ] we have then f o r any n^N, L e t t i n g u(E~UWi n ~UwJ ^ b + 3£ . i ^ K ' i>K 1 D n = E ~ I J W± ~UWi , i < K ' 1 , n i>K and a p p l y i n g theorems 2 0 . 6 and 4 . 2 we have K u ( E ~ U B i ~ U W i ) = u(UDn) = l i m U D n ^ b + 3e i=o i>K y new n-xx/ whence ^ > M u(E~UBi~UWi) ^ h + 36 . i=o i>K Since f o r I > K we have W^€ U , we conclude K y"u(E~ U B i ) ^ b + 3& +^uUWi i=o 1>K 4 b .+ 3£ + Zl^Wi < h + 56 , i>K and so M T T ( E ~ U B i ) < b,+ 5e . u lew x Taking the supremum over U £ ^  and l e t t i n g 6->0,, we have u ( E ~ U B i ) ^ b-lew Proof of 2 0 . 7 : Let a = l i m U yA n and suppose a<oo n Choose s e t s P ( i , n ) such t h a t f o r i £ w , 0 < n £ w , i ) A n c U p ( ^ n ) ; lew i i ) r P ( i , n ) > r P ( i + l , n ) ; i i i ) ^v?{i.?n) < M T J A n + l / n ; and lew i v ) P ( i,n)£U*. 109 In connection w i t h i i ) , we note t h a t f o r any P(i,n) w i t h * P ( i , n ) = 0 , by d e f i n i t i o n 20.1.3 and the f a c t t h a t the space i s Hausdorff, P ( i , n ) i s a s i n g l e t o n or i s empty, and has jUy.and ju-measure zero. Thus the countable f a m i l y of a l l such P ( i , n ) a l s o has measure zero and so we may without, l o s s of g e n e r a l i t y asBume that f o r every IEto a n d n G u ) , r P ( i , n ) > 0 . Then f o r any nGto, only a ' f i n i t e number of the P(i, n ) can have the same T - v a l u e and we can c a r r y out t h e o r d e r i n g by non-i n c r e a s i n g r-values.. Prom i i i ) we have v), l i m ]>]-&P(i,n) = a> n new We now apply a d i a g o n a l process t o our s e t s P ( i , n ) . Let ( f ^ , <x.GD) be a u n i v e r s a l subnet, of the net (n,nGto) (see K e l l e y , problem 2 J ( d ) ) . Then from i ) and the f a c t t h a t A i s an ascending sequence we have a) For each nGCu there e x i s t s j 3 n G D such t h a t f o r a l l «-»An> A n C i J P ( i ^ a ) , iew and from v ) , b) , l i m X > P ( i , f a ) = a . <* lew C l e a r l y f o r a l l l G w , otG D, c) . r P ^ , ^ ) > T P U + l , ^ ) ; d) P ( i , f o t ) G U*. For each i G w , by theorem.19.8 there i s a Vj_G ^  which i s a l i m i t p o i n t of the sequence ( P ( i , n ) , n£u>). Then I s a l i m i t p o i n t of the subnet ( P ( i , f o t ) , <x6 D) and s i n c e ( f ^cGGD) i s u n i v e r s a l , by problem 2 J , (b) and ( a ) , of K e l l e y , e) Vi = (fe-lim P(i,foc) f o r each i € w . oc Now choose symmetric ME*U, WoWCV. By lemma 20.12 f o r each IGO) there e x i s t s B i such t h a t zB± = tV±, V i C B i C W t V j J , and s i n c e the Bj_ are independent of n, f o r every ngo), 2 > V ± + u j ( A n ~ U B i ) l i m 2>P(i,f„c).=-a. lew iew ct ieco I t f o l l o w s from theorems 20.6 and 4.2 t h a t X > V ± + u d j A n - U B i X a . ieco 7 new lew By d), e) and lemma 19.9, V 1XV 1CtJCWoUoW and so by 3•11.5, B 1 X B 1 C W . [ V 1 ] X W [ V 1 ] C W © W o U o W o W G V o U o V f o r 16 CO and B ± e (VoUoV)* f o r iGco. Hence > W V U A n < > L V o U o V ( ( j A n - U B j . ) +.M.VOUOV U B l new new lew , Jew 1 ^y^Vouov O n ~ U B ± ) + new iew lew ^ / t ( U A n ~ U B i ) + , I t 7 1 4 a . ' n P A i i^w ie=w 1 ew e i w21. Sets of non- (T- f i n i t e measure. In t h i s s e c t i o n we d e f i n e a p a r t i a l order >- on the f u n c t i o n s i n T and examine some consequences of t h i s order-i n g . The d e f i n i t i o n extends t h a t i n 6.1 of Sion and Sjerve I l l t o compact Hausdorff spaces. Again we assume (X,^) i s a compact Hausdorff space, "U i s the u n i f o r m i t y f o r X which induces J), J i s the f a m i l y of subsets of X. For any gauge r on >3 , JJL or jx- i s the measure of chapter I I I generated by H a n d X . 2 1 . 1 DEFINITION. Suppose r x , ^ G T , Then r x y X2 i f f g i v e n £ > 0 , there e x i s t s U G ' U s u c h t h a t i f A X A C U , then T±A ^ £ X2A 2 1 . 2 DEFINITION. I f ( X , ^ ) , i s a t o p o l o g i c a l space, then A> i s a n a l y t i c In X i f f A = f[oc] f o r some ocGK^j and some continuous f u n c t i o n f on oc t o X, where K* i s the f a m i l y of c l o s e d , compact s e t s i n a t o p o l o g i c a l space ( x ' , $ ) . (For a good resume of the theory of a n a l y t i c s e t s , see B r e s s l e r and Sion [ 4 ] . ) In c o n n e c t i o n w i t h these concepts, Sion and Sje r v e proved the f o l l o w i n g theorems. THEOREM (6.4). Suppose f o r every nGw, t r ) + 1>r i ie T, and (6 i s the f a m i l y of a l l s e t s of the form L J ^ , where , new y^n) - o f o r nGu>. I f E. i s a n a l y t i c i n X and E ^  <£>, then there e x i s t s ZE T such t h a t Tl>-'Cn f o r nGto, and E has non-(T- f i n i t e ju(' c)-measure. THEOREM. (6.5) . Suppose t £ T, E i s a n a l y t i c i n X and has non-(T- f i n i t e j A ^ ^ - m e a s u r e . Then there e x i s t s a -t1 G T such t h a t x, >- % , and E has non-<r- f i n i t e u.^-measure. 1.12 THEOREM ( 6 . 6 ) Suppose t G T , E i s a n a l y t i c i n X and (x) has non-<T- f i n i t e u> '-measure. Then t h e r e e x i s t s a compact R (1) C C E , such t h a t C has non-(T- f i n i t e JJ~ -measure. The f o l l o w i n g theorem, along w i t h lemma 6 . 2 i n Sion and S j e r v e , shows t h a t the e x i s t e n c e of f u n c t i o n s i n T ordered by >- i s e q u i v a l e n t to m e t r i z a b i l i t y i n a compact Hausdorff space.. 2 1 . 3 THEOREM. I f there e x i s t X±, Tg.G T such that r - ^ V t g , then (X, Jf) i s m e t r i z a b l e . 2 1 . 4 REMARK. By the above theorem, the h y p o t h e s i s i n theorem ( 6 . 4 ) of Sion and S j e r v e i m p l i e s that the space i s m e t r i c , f o r which case t h a t theorem was proved. However, i f the compact Hausdorff space Is non-metrizable then theorem 2 1 . 3 shows t h a t theorem ( 6 . 5 ) i s f a l s e i n t h i s space. The q u e s t i o n remains of whether theorem ( 6 . 6 ) can be g e n e r a l i z e d to the compact Hausdorff case. An e s s e n t i a l step i n the p r o o f of t h a t theorem i n a m e t r i c space was the con-s t r u c t i o n , given T^G T, of. a f u n c t i o n TTgG T such t h a t % 2>- . T h i s cannot be done i n a non-metrizable space, as theorem 2 1 . 3 shows, so t h a t the p r o o f does not g e n e r a l i z e . The f o l l o w i n g theorem and example are r e l a t e d t o the q u e s t i o n . 2 1 . 5 THEOREM. Suppose TGT. I f X . i s of < r - f i n i t e LL-measure, then X i s m e t r i z a b l e . 1.13 21.6 REMARK. The a t t r a c t i v e h y p o t h e s i s t h a t every non-m e t r i z a b l e a n a l y t i c subset of a compact Hausdorff space con-t a i n s a non-metrizable compact subset i s f a l s e (see c o n c l u d i n g example). PROOFS AND EXAMPLE 21.7 LEMMA. I f there e x i s t s ; a descending sequence U of symmetric members of It c l o s e d ,In the product topology on XXX, and T G T such t h a t f o r each nGco, A X A C U => T A < l / n , then l^ jj n e w l s a base f o r fy and ( X , £ ) . Is m e t r i z a b l e . Proof: Since U i s symmetric f o r each nGw, P)U I s . neco symmetric. Suppose n€w' Then there e x i s t s A C X such t h a t AXACjl A and A X A C f a n -ned) Since A X A C U n f o r each ngui, we have rA = 0. On the other hand, s i n c e TGT and A X A (£ A , we have rA > 0 by d e f i n i t i o n 20.1.3. We conclude t h a t Now s i n c e the space i s compact, It c o n s i s t s of a l l neighborhoods of A . Let JJGU. Then there e x i s t s G, open i n XXX, such t h a t A C G C U. But A i s the i n t e r s e c t i o n of a descending sequence of s e t s 114 U n c l o s e d i n the product topology on XXX, which i s compact. Hence f o r some nEfc), U n C G C U . Hence {u^] i s a base f o r Ii and by theorem 6.13 of K e l l e y , n€w (X, &) i s m e t r i z a b l e . Proof of 21.3: Suppose T^, TgE T, t >-Tg. By d e f i n i -t i o n 20.1.5 there e x i s t s K such, t h a t f o r a l l AGX, T 2 A<K.. Usin g d e f i n i t i o n 21.1, choose f o r each n ^ , 1 , c l o s e d symmetric U n E Ii (the c l o s e d symmetric members of H form a base f o r U (3.11.4)) such t h a t . U n + 1 C U n , and A X A C U n => r xA ^ ^ _ . r 2 A ^ l / n . By, lemma 21.7, (X, >#).is m e t r i z a b l e . Proof of 21.5: Suppose X i s of (T-finiteju-measure. i ) : There e x i s t only countably many p o i n t s x E X such t h a t t { x } > 0 . Although s i n g l e t o n s may not be ji-measurable, s i n c e the space i s Hausdorff, any two p o i n t s x, y can be separated by d i s j o i n t c l o s e d ^ s e t s , and s i n c e c l o s e d JS^ s e t s are jx-measurable (theorem 18.3.2), /x{x, y} = JLA[X} + jx{y]. Since X has <T-finite measure, th e r e can be only a countable number of p o i n t s x w i t h JJ.{X] >0. But JJ.{x] = z{x} . i i ) L e t P = | x E X : z{x] = o]. We show t h a t F i s c l o s e d and hence compact. I f (x^jOCED) i s a net i n F•which converges t o y E X i n the topology , then the net ( { X ^ ^ C C E D ) i n J converges to {y} i n the topology k.. By c o n t i n u i t y of X i n k , r [ x j = 0 f o r ccGD =>T{y} = 0 and so y £ F . Hence F i s c l o s e d . i i i ) . F. w i t h the r e l a t i v e topology Jf^ i s m e t r i z a b l e . L e t V be the r e l a t i v i z a t i o n of H t o F, so Jj^ = Ty (see K e l l e y , p. 1 8 2 ) . Then f o r every £ > 0 there e x i s t s U G ^ such t h a t B X B C U =>xB < e , f o r otherwise suppose t h a t f o r each UGV, there, e x i s t s A ^ C P such t h a t A y X A u C U and ZAJJ > e. Then, s i n c e F i s compact, given the net (A^, JJG1/), by theorem 1 9 . 8 there i s a subnet ( A a , O C G D ) and A G F such t h a t A = (A.-lim Aoc. oc By the c o n t i n u i t y of X, XA = l i m rA^ > £. oc A l s o A X A G A , f o r l e t U G V and choose symmetric V£'Y such t h a t VoVoVCU. By d e f i n i t i o n 1 9 . 7 there e x i s t s T G D such t h a t f o r p » Y , A G V [ Ap ] . Since A y X A ^ . C U, f or each U G V and (A f t, oCG D) i s a subnet of (Arj, U G f y ) we can and do choose p»Y such t h a t Ap X A J J C V . Then by 3 . 1 1 . 5 , A X A C V [ A p ] X V [ A ^ ]GVoVoVGU,. and so A X A G U f o r every U G ^ . Hence A X A C A . T h e r e f o r e A = {x} f o r some x g F and so TA = 0, c o n t r a d i c t i n g r A > £ > 0. We conclude t h a t given £ > 0 , there e x i s t s U G ^ such t h a t f o r A X A C U , rA < £ . Now choose a sequence U. i n If such t h a t f o r 0 < n £ t o , U n i s c l o s e d and symmetric, F n + l C U n , and A X A C U n => Vk^-l/n . Then by lemma 2 1 . 7 , (F, j ^ p ) , i s m e t r i z a b l e . We note that (F, « £ p ) i s separable s i n c e i t . i s compact. We have then t h a t X i s a compact Hausdorff space which i s a union of countably many separable m e t r i z a b l e spaces (F and [xj f o r each x G X ~ P ) . Hence by a theorem of Smirnov (see Stone [ 1 5 ] , p r o p o s i t i o n (B)), X i s m e t r i z a b l e . Example f o r 2 1 . 6 : T h i s example was suggested by t h a t i n the remark f o l l o w i n g c o r o l l a r y 2 i n Stone [ 1 5 ] . Let X = { ( 0 , 0 ) } U {(x,y)€ R X R : x OJ. For (x, y) / (0, 0) take as a neighborhood system the neigh-borhood system of (x,y) i n the plane r e l a t i v i z e d t o X. For (0,0) take as a base f o r the neighborhood system- the f a m i l y of open s e t s i n the u s u a l topology on the plane which c o n t a i n the y - a x i s . Let & -be the r e s u l t i n g topology on X. Then 1 ) Jf i s c l e a r l y Hausdorff. i i ) (X, Jr) i s not m e t r i z a b l e , s i n c e Jt does not s a t i s f y the f i r s t . a x i o m of c o u n t a b l l i t y . Given any countable f a m i l y of open s e t s i n the plane which c o n t a i n the y - a x i s , we can c o n s t r u c t an open s e t c o n t a i n i n g , t h e y - a x i s which excludes 117 p o i n t s of every member of the countable f a m i l y , i . e . the neigh-borhood system of (0,0) i n J) does not have a countable base. i l l ) • I f A C X i s compact, then A i s m e t r i z a b l e . X i s the union of a countable number of separable m e t r i z a b l e spaces, and so the same i s t r u e of any A C X . But s i n c e A i s compact, i t i s m e t r i z a b l e by the theorem of Smirnov (Stone [ 1 5 ] , p r o p o s i t i o n (B.)). i v ) X i s a n a l y t i c s i n c e i t i s the union of a countable number of compact s e t s (see 2 1 . 2 ) . v) (X,& ) i s normal, hence completely r e g u l a r s i n c e i t i s Hausdorff, and so by theorem 5.15 i n K e l l e y , can be embedded i n a compact Hausdorff space. To see t h a t i t i s normal, con-2 s i d e r the map f : R —> X d e f i n e d by f ( ( x , y ) ) = ( ( x , y ) i f x / 0. \ (0,0) i f x = 0. The v e r i f i c a t i o n t h a t f i s continuous i s t r i v i a l . I t i s a l s o 2 easy to see t h a t the image of an open set i n R c o n t a i n i n g the y - a x i s i s open, and the image of an open - set not i n t e r s e c t i n g the y - a x i s i s open. Let A, B be c l o s e d i n J) , A f ) B = 0 . I f (0,0) i s a member of one of them, say A, then f - 1 [ A ] , f _ 1 [ B ] 2 - 1 are d i s j o i n t c l o s e d s e t s i n R and f [A] c o n t a i n s the y - a x i s . 2 2 Since R i s normal, there e x i s t open s e t s C, G i n R such that f - 1 [ A ] C C , f _ 1 [ B ] C G > and C f l G =0. Now C c o n t a i n s the y - a x i s , so f [ C ] and f [ G ] are open, d i s j o i n t and A C f [ C ] , B C f [ G ] . 118 I f ( 0 , 0 ) i s a member of n e i t h e r A nor B, then f _ 1 [ A ] , — 1 2 f [B] and the y - a x i s are d i s j o i n t c l o s e d s e t s i n R and we p can f i n d d i s j o i n t open s e t s C and G i n R which do not i n t e r -s e c t the y - a x i s and which c o n t a i n r e s p e c t i v e l y f - 1 [ A ] and f"1.[B]. But then f [ C ] are f [ G ] and. open, d i s j o i n t and A C f [ C ] , B C f [ G]. BIBLIOGRAPHY 119 1. A. S . B e s i c o v i t c h , On the fundamental p r o p e r t i e s of l i n e a r l y measurable plane s e t s of p o i n t s , Math. Annalen 9« (192«j, 4 2 2 ^ 6 ^ 7 2. W. W. Bledsoe and A. P. Morse, _A t o p o l o g i c a l measure c o n s t r u c t i o n , P a c i f i c J . Math. 13 ( 1 9 6 3 ) , 1 0 6 7 - 1 0 8 4 . 3. N. Bourbaki, Elements de mathematiques, I I , p a r t I, v o l . I l l , T o p o l o g i e ggne*rale, Hermann, P a r i s , 1 9 6 1 . 4. D. W.- B r e s s l e r and M. Sion, The c u r r e n t theory of a n a l y t i c s e t s , Canad. J . Math. l 6 ^ T 9 6 4 ) , 2 0 7 - 2 3 0 . 5 . ' C. Carathe'odory, tiber das l i n e a r e Mass von Punktmengen— ein e Verallgemeinerung des L S n g e n b e g r i f f s , Nachrichten Ges. d. Wiss. Go'ttingen, 1914, 404-426. 6. P. Hausdorff, Dimension und ausseres Mass, Math. Annalen . 79 ( 1 9 1 9 ) , 1 5 7 - 1 7 9 . 7. J . L. K e l l e y , General topology, Van Nostrand, P r i n c e t o n , 1959. 8 . E. Michael, T o p o l o g i e s on spaces of subsets, Trans. Amer. Math. Soc. 7 1 ~ T l 9 5 1 ) , 152-T82 - ; 9 . M. E. Monroe, I n t r o d u c t i o n t o measure and i n t e g r a t i o n , Addison-Wesley, Reading, 1959 . 1 0 . W. J . P e r v i n , U n i f o r m i z a t i o n of neighborhood axioms, Math. Annalen 147 ( 1 9 b 2 ) , 3 1 3 - 3 1 5 . 11 . , Q u a s i - u n i f o r m i z a t i o n of t o p o l o g i c a l spaces, . Math. Annalen: 147 ( 1 9 b 2 ) , 3 l6=3" l7 . 12 . C. A. Rogers and M. Sion, On Hausdorff measures i n t o p o l o g i c a l spaces, Monat. f u r Math. 67 (196~3T, 2 b 9 - 2 7 » . 1 3 . M. Sion and D. S j e r v e , Approximation p r o p e r t i e s of measures generated by continuous set f u n c t i o n s , Mathematika 9 1 1 9 b 2 ) 7 145-146. ' 14. A. H. Stone, Paracompactness and product spaces, B u l l . Amer. Math. Soc. 5^ U 9 4 « ) , 9 7 7 - 9 » 2 . 1 5 . —• , M e t r i z a b i l i t y of unions of spaces, Proc. Amer. Math. Soc. 10 ( 1 9 5 9 ) , 3bl-36"5-1 6 . J . W. Tukey, Convergence and u n i f o r m i t y i n topology, Annals of Mathematical S t u d i e s , no. 2 , P r i n c e t o n , 1 9 4 0 . 

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