NON-STANDARD ANALYSIS by GLEN RUSSELL COOPER B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of ' Mathematics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1975 In presenting an advanced the I for shall agree scholarly by his of this written thesis in at University the make that it purposes thesis for partial freely permission may representatives. be It financial for 8, of British Canada of Columbia, British by gain Columbia for the understood of University Vancouver of extensive granted is fulfilment available permission. Department The degree Library further this shall Head be requirements reference copying that not the of agree and of my I this or allowed without that study. thesis Department copying for or publication my ii Supervisor: Dr. P. Greenwood ABSTRACT In t h i s t h e s i s some c l a s s i c a l theorems of a n a l y s i s are provided w i t h non-standard proofs. In Chapter 1 some compactness theorems are examined.. the monad u(p) of any point to a family H p contained i n a set X of subsets of X ) i s defined. (and r e l a t i v e Using monads, a non- standard c h a r a c t e r i z a t i o n of compact f a m i l i e s of subsets of X In 1.2 i t i s shown that the monad to H ) T(H) remains unchanged i f H on X containing In 1.1 u(p) of any point p e X i s given. (relative i s extended to the smallest topology H . Then, as an immediate consequence, the Alexander Subbase theorem i s proved. In 1.3 monads are examined i n t o p o l o g i c a l products of t o p o l o g i c a l spaces. Then, i n 1.4 and 1.5 r e s p e c t i v e l y , both Tychonoff's theorem and Alaoglu's theorem are e a s i l y proved. In Chapter 2 various extension r e s u l t s of T a r s k i and Nikodym ( i n the theory of Boolean algebras) are presented w i t h rather short proofs. A l s o , a r e s u l t about Boolean covers i s proved. The techniques of non-standard a n a l y s i s contained i n Abraham Robinson's book [Robinson 1974] are used throughout. The remarks at the end of each chapter set f o r t h p e r t i n e n t references. iii TABLE OF CONTENTS . Page Introduction Chapter 1 Chapter 2 Bibliography 1 1.1 Compact Families of Sets 4, 1.2 Alexander Subbase Theorem . 5 1.3 Products of Topological Spaces 7 1.4 Tychonoff's Theorem 8 1.5 Alaoglu's Theorem 9 1.6 Remarks 10 2.1 Tarski's Result 11 2.2 Nikodym's Result 13 2.3 Boolean Covers 14 2.4 Remarks 16 18 iv ACKNOWLEDGEMENT S I wish to thank my advisor, P r i s c i l l a Greenwood, f o r proposing t h i s t h e s i s and guiding i t s development. Andrew Adler f o r h i s kind c r i t i c i s m . As w e l l , I wish to thank F i n a l l y , I wish to thank the Department f o r providing f i n a n c i a l assistance. This t h e s i s i s dedicated to my w i f e , Adriana (nee Buonanno) Cooper. i INTRODUCTION Some techniques of non-standard a n a l y s i s contained i n Abraham Robinson's book [Robinson 1974] are reviewed here. R e c a l l that i f s t r u c t u r e on a set X { X } on a s e t X ^ ^ i X T If such that { X } extends {X } T contains S . (in particular X X ) and such that the following p r o p e r t i e s are s a t i s f i e d . i s a higher order sentence i n {X } and interpretation in f u l l , normal higher order then there e x i s t s a normal higher order s t r u c t u r e T X t s n e S is its T r* in { X } X then S * i s true i n {X } i f f S • i s true X { X } . This property i s sometimes c a l l e d the t r a n s f e r For the next property we r e c a l l some d e f i n i t i o n s . Suppose principle. Q(x,y) i s a binary r e l a t i o n i n {X^} . The domain of Q.', domain (Q) , i s defined as the set of a l l . x in i n {X } such that Q(x,y) f o r some y {X^_} . The range of Q , range (Q) , i s defined s i m i l a r l y . binary r e l a t i o n , x^,...,x Q(x^,y) Q(x,y) i n {X } i s c a l l e d concurrent e domain (Q) there e x i s t s some f o r K = l , . . . , n . I f Q(x,y) i f whenever y e range (Q) such that i s a concurrent, r e l a t i o n i n {X } then there e x i s t s some .x A binary y„ i n { X } •'Q x ( c a l l e d an . * i d e a l element f o r Q ) such that Q(x,yp) i s true i n { X^_} f o r x e domain (Q An enlargement of the f u l l , normal higher order s t r u c t u r e {X^.} on a set X i s any normal higher order s t r u c t u r e s a t i s f y i n g the above p r o p e r t i e s . { X^_} on a set X Thus, by the above, any f u l l , normal higher order s t r u c t u r e possesses and enlargement (which therefore s a t i s f i e s the t r a n s f e r p r i n c i p l e and assigns i d e a l elements to concurrent binary r e l a t i o n s ) . Suppose a normal higher order s t r u c t u r e {. j on a set i s an enlargement of the f u l l , normal higher order s t r u c t u r e on a set X . R e c a l l that, i n general, { X^} AA Thus i f { AA then, f o r each x , need not be f u l l . A i n some X A X^ - AA contained {X^_} the f u l l , normal higher order s t r u c t u r e on 1 S X X^ need not be empty. Any r e l a t i o n A X - T X i s c a l l e d e x t e r n a l while the r e l a t i o n s T A contained in X are c a l l e d i n t e r n a l . Any r e l a t i o n contained i n T some X i s c a l l e d standard. T Some general properties of enlargements are r e c a l l e d here. A ft Suppose the normal higher order s t r u c t u r e { X^} on the set X i s an enlargement f o r the f u l l , normal higher order s t r u c t u r e the set X . Let A {X •}• . Then A U B and B be sets (that i s , unary r e l a t i o n s ) i n i s contained i n {X } since T A {X } i s f u l l . X A A, AL) B Let X A B {X^} on A and (A U B) be the extensions i n { X } of A, B and ,x respectively. Then * ( A U B) = *A U *B . S i m i l a r r e s u l t s hold for a l l f i n i t e Boolean combinations of sets. A A Let { R } be a normal higher order s t r u c t u r e on a set R x . ° which i s an enlargement of the f u l l , normal higher order s t r u c t u r e {R } on the set R of r e a l numbers. R e c a l l t h a t , by t r a n s f e r , the T 1 sentence i n {R } which states that x R i s an ordered f i e l d under the A A usual r e l a t i o n s also states (when i n t e r p r e t e d i n { R J ) T that R is A an ordered f i e l d under the same r e l a t i o n s (when extended i n { R }) . x A A Thus R ( s t r i c t l y speaking { R^} ) may be c a l l e d a f i e l d of non-standard r e a l numbers. Let be the absolute value function i n { R } . ii A R e c a l l that a non-standard r e a l number x e R T i s called f i n i t e i f 3 A. . |x| <_ y f o r some y e R (otherwise i t i s c a l l e d i n f i n i t e ) . Also, A a non-standard r e a l number |x| < y x e for a l l positive But the binary r e l a t i o n R i s called infinitesimal i f y e R . Clearly Q(x,y) in o e R meaning i s infinitesimal. o < y < x is A concurrent. Thus there e x i s t s some for a l l p o s i t i v e x e R . Hence y^ y^ i n { R^_} such that Q(x,y^) i s a non-zero i n f i n i t e s i m a l A non-standard r e a l number. Since R contains non-zero i n f i n i t e s i m a l s A i t contains i n f i n i t e numbers. From t h i s i t i s easy to show that R A i s non-archimedean. R e c a l l that i f x e R i s a f i n i t e non-standard r e a l number then there e x i s t s a unique standard r e a l number (called the standard part of x ) such that x - °x °x e R i s infinitesimal (write x - °x and c a l l x i n f i n i t e l y close to °x ) . I t i s easy ^ o o o to show that i f x , y e R are f i n i t e then (x+y) = x + y and °(xy.) = °x°y . Obviously x = °x i f x e R . I f x e R l e t the A monad u(x) of satisfying x be defined as the set of a l l f i n i t e y - x . R e c a l l that a set A C R ye R of standard r e a l numbers A i s open i f f whenever x e A then u(x) C A . Hence a set B C R of standard r e a l numbers i s . c l o s e d i f f whenever x e.R and A y(x) 0 B i cf> then x e B . Note that a set A C R numbers i s f i n i t e i f f there e x i s t s a b i s e c t i o n from of standard r e a l {1,...,n} onto A A ( f o r some n e N ) . An i n t e r n a l set A C numbers i s c a l l e d {1,...,(JJ} of non-standard r e a l ^ - f i n i t e i f f there e x i s t s an i n t e r n a l A from R onto bisection A A ( f o r some co e e x i s t s f o r ^ - f i n i t e i n t e r n a l sequences. N ) . A similar definition 4 F i n a l l y , r e c a l l that i t may be assumed that the f u l l , higher order s t r u c t u r e {X } on a set X contains the set R of standard T r e a l numbers (extend X enlargement of contains a set {X_^} to X U R From now on such enlargements i f necessary) and thus any R of non-standard r e a l numbers. are assumed f i x e d i n any d i s c u s s i o n . CHAPTER 1 1.1 Compact Families of Sets definition Let X be a set and l e t S(X) be the power set of X and let H C S ( X ) be a set of subsets of X . I f x e X then ft y(x) = Pi { A : x e A and A e H} i s c a l l e d the monad of x ft ( r e l a t i v e to H ) . A point to H ) i f p e u(x) f o r some H )) . A set X members of on X p e H then i s called X i s c a l l e d near-standard x e X (write p - x ( r e l a t i v e to H-compact i f every cover of has a f i n i t e sub-cover. (relative Note that i f H X by i s a topology H-compactness i s j u s t compactness i n the sense of topology. Lemma Let X be a set and l e t H be a set of subsets of X . ft Then X i s H-compact i f f each p e X i s near-standard ( r e l a t i v e to H ) . Proof ft Suppose X i s H-compact but assume some p e X i s not 5 near-standard ( r e l a t i v e to H ) . Then f o r each some A e H x {A : x e X} x (A U * x eA covers so there e x i s t s some f i n i t e set • {•x, ,. . . ,x } C X I n X '= i there e x i s t s such that X X 1 • ' "U A X-, so p e A X 1 n '"LJA . Thus x n 1 A U However T X = A U x.. such that "•L)A ) = X-, . x but p 4 A x x e X ' T7 n K K = 1 , . . . , n . This c o n t r a d i c t i o n implies that ( r e l a t i v e to H ) . Thus each f o r some X . p e X p i s near-standard i s near-standard ( r e l a t i v e to H ). Suppose, conversely, each to H ) but assume {A } C H covering X X p e X i s not H-compact. i s near-standard (relative Then there e x i s t s some but possessing no f i n i t e sub-cover. Thus the binary r e l a t i o n Q(A^,x) meaning x \. A^ i s concurrent so there e x i s t s some x^ e X such that x^ 4 A f o r a l l a . However Q Q a x^ - x ( r e l a t i v e to H ) f o r some x e X . But x e A f o r some a . Q a * Thus x„ e A This c o n t r a d i c t i o n implies that X i s H-compact. Q a T 1.2 Alexander Subbase Theorem definition • Let X > be a set and l e t H C S ( X ) be a set of subsets of X . Let x ( H ) denote the smallest topology on R e c a l l that T ( H ) = H ' U {X} where X containing H . H ' i s the set of a l l unions of f i n i t e i n t e r s e c t i o n s of members of H . Note that i f x e A and A e H ' then x e A, C\ ••• f l A C A •1 n f o r some f i n i t e set {A,,...,A } C H I n 6 Theorem Let X be a set and l e t H be a set of subsets of be the smallest topology on p e X and x e X . Then ( r e l a t i v e to X containing p - x X and l e t x(H) H . Furthermore, l e t ( r e l a t i v e to H ) iffp - x x(H) ) . Proof Suppose p - x p - x ( r e l a t i v e to ( r e l a t i v e to H ) . I t s u f f i c e s to show that x(H) ) . Suppose s u f f i c e s to show that ft p e A . assume A e h" A ^ X . Then x e A If A = X where f i n i t e i n t e r s e c t i o n s of members of H' and then A e x(H) . I t p e * A . Thus, i s the set of a l l unions of H . Thus x e A,H ' " O A for some f i n i t e set {A^ C A n x e A^ 1 A ^ } C H . In p a r t i c u l a r , ft for • K = 1,...,n . Hence p e ( r e l a t i v e to H ) . But then so p e ft A A^ f o r K = 1,...,n since p - x p e *A. f] ' ' ' 0 *A = * (A 0 • • • 0 A ) C *A 1 n 1 n n as desired. The reverse i m p l i c a t i o n i s c l e a r . Corollary Let X (Alexander Subbase) be a set and l e t H be the smallest topology on then X be a set of subsets of X containing H . X If X and l e t i s H-compact i s x(H)-compact. Proof ft Suppose X i s H-compact. Let p e x(H) X . By 1.1 there e x i s t s some x e X such that i t s u f f i c e s to show that p - x p - x ( r e l a t i v e to H ) . T(H) ) . ( r e l a t i v e to By 1.1 But t h i s follows from the above theorem. 1.3 Products of Topological Spaces definition If (where x_^ {(X^,T_^) : i e 1} i s a family of t o p o l o g i c a l spaces i s the topology on product (where IIx X^) let (ITX^, IIT_) be i t s t o p o l o g i c a l i s the product topology on the product topology on IIX ) . IIX^ i s the smallest topology on R e c a l l that IIX containing a l l sets of the form A = {f e nx. : f ( i ) e A. } x X x ± where i e I and A. X ex. l 1 . Hence the product topology on IIX. i s 1 1 the set of a l l unions of sets of the form A = where i..,...,i 1 n O K=l e l n {f e nx. : f ( i ) e A. } K 1 and A. K X k ex. K X for K = l,...,n . X Lemma Let {(X_^,x^) : i e 1} be a family of t o p o l o g i c a l spaces and l e t ft (nx^nxj be i t s t o p o l o g i c a l product. h e IIX . Then i g - h ( r e l a t i v e to Suppose Tlx ) g e (rOL) i f f g(i) - h(i) and (relative to • x ) for i e l Proof Suppose g - h s u f f i c e s to show that ( r e l a t i v e to g(i-,) - - h ( i - ) ) . Let i ^ e I . I t ( r e l a t i v e to x. ) . Let * h ( i ) e A. 1 and A. l 1 But ex. l X . I t s u f f i c e s to show that X X h e {f e nx. : f ( i j e A. } and x 1 x^ Hence g e {f e nx_^ : f ( i ^ ) e A_^ } ft 1 g ( i . ) , e A. as desired. 1 Thus g(i) e A. l {f e nx. : f ( i , )• e A. } e nx. x 1 x^ x since g - h ( r e l a t i v e to nx_^ 1 Suppose, conversely, g ( i ) - h ( i ) l e i . I t s u f f i c e s to show that Let * ' g e .A . But i ,...,i e I h e A x^ ) f o r ( r e l a t i v e to nx^ ) . A e nx. . Then i t s u f f i c e s to show that x £ h e J {f e HX. : f(i„) e A. } C A f o r some K—X X K i ' K and some A. e x . f o r K = l , . . . , n . In p a r t i c u l a r and r h ( i ) e A. and iv x^. ( r e l a t i v e to g - h ( r e l a t i v e to A. x x. ) K R ex. x f o r K = l , . . . , n . But R for K = l,...,n so X g(i„) - h ( i ) K K 7r g ( i ) e A. f o r V K ft K = l , . . . , n . Hence g e {f e JJX. : f ( i ) e A. } K 1 Therefore, n g e H ' { f e nx. : f ( i J K—X X K A ft 11 * 0 {f T nx. : f ( i ) e A. } C x x K 1 K 1.4 Tychonoff's Theorem e Theorem (Tychonoff) K for K = l,...,n X e A. } = Xj^ A . Thus ft g e A as desired, Let { ( X ^ x ^ ) : i e 1} (ITX^IIT^) for be a family of t o p o l o g i c a l spaces and l e t be i t s t o p o l o g i c a l product. Suppose X_^ i s x^-compact IT.X. i s Tlx . -compact. x x i e I . Then Proof' A. . ' Suppose i s x^-compact f o r i £ I . Let g X^ By 1.1 i t s u f f i c e s to show that g E (IIX^) . i s near-standard ( r e l a t i v e to A II ) . But ( r e l a t i v e to g(i) e f o r i e I . By 1.1, g ( i ) x. ) since X. i s x.-compact, x x x Thus Let h(i) = x ( r e l a t i v e to Thus 1-5 g g ( i ) - x. " 1 ( r e l a t i v e to f o r i E I . Then ± x. ) x h E -IIX i s near-standard ( r e l a t i v e to f o r some and x_^) f o r i e I . By 1.3, g - h IIx^ ) i s near-standard x. e X. . x x g(i) - h ( i ) ( r e l a t i v e to JIx^) . as desired. Alaoglu's Theorem definition Let X be a r e a l normed l i n e a r space and l e t L ( X ) be the set of bounded, l i n e a r f u n c t i o n a l s on f e L(X) implies norms on X and Cartesian power X X . R e c a l l that | | f ( x ) | | <_ | | f | | | |x| | L ( X ) ) . Note that (where || || x e X and denotes the L ( X ) i s contained i n the X . Let x^ be the topology of pointwise convergence * y X . By 1.3, i f g E (K ) and h e R then g - h ( r e l a t i v e on R to x^ ) R i f f g(x) - h(x) f o r X E X . Let B = {f E L ( X ) : ||f||£-l}. 10 Theorem (Alaoglu) B is x -compact P Proof Let g e ( r e l a t i v e to ,*llg(x)II 1 * II g I I — x ) )• • x x x E X . g(x) - h(x) 1.7 By 1.1 i t s u f f i c e s to show that * f o r some h e B . I f x E X then *l|g|1*1 M | £ *| M I|g( )I I 1 if P B . 111 I l lI x Let | (since ge*(L(X)) particular, i f x £ X • Hence g(x) for x E X . then Then Thus, by 1.3, g - h and | | x | | = | | x | | so i s f i n i t e and h(x) = °g(x) f o r x E X . g - h °g(x) i s defined h E B . ( r e l a t i v e to But x^ ) . Remarks In 1.1 a g e n e r a l i z a t i o n i s provided f o r Robinson's non-standard c h a r a c t e r i z a t i o n of compactness i n t o p o l o g i c a l spaces [Robinson 1974]. This i s done by observing that the d e f i n i t i o n of compactness, as w e l l as Robinson's c h a r a c t e r i z a t i o n of i t , does not require the axioms f o r a t o p o l o g i c a l space. Then, i n 1.2, t h i s g e n e r a l i z a t i o n y i e l d s the Alexander subbase theorem. In 1.3 the near-standard r e l a t i o n i n t o p o l o g i c a l products i s reduced to one i n v o l v i n g the coordinate spaces. Then, i n 1.4, Tychonoff's theorem i s e a s i l y proved. appear i n a book of Robinson [Robinson 1974]. may be seen elsewhere, however. proved i n 1.5. Both 1.3 and 1.4 The usefulness of 1.3 For instance, Alaoglu's theorem i s This i s p o s s i b l e since Alaoglu's theorem deals with 11 compactness r e l a t i v e to a topology of pointwise convergence, and thus r e l a t i v e to a product topology. One might observe that c l a s s i c a l proofs of the above r e s u l t s follow similar l i n e s . Hence the Alexander Subbase theorem provides an immediate proof of Tychonoff's theorem [Kelley 1955]. But Tychonoff's theorem y i e l d s Alaoglu's theorem by providing a compact t o p o l o g i c a l product i n which the unit b a l l of the dual of a normed l i n e a r space e x i s t s as a closed (hence compact) subspace [Bachman-Narici 1966]. Thus i n the standard proof of Alaoglu's theorem some a n a l y s i s i s required to show that the above subspace i s closed (hence compact). The lemma i n 1.3 eliminates t h i s requirement by providing compactness directly. CHAPTER 2 2.1 Tarski's Result definition Let theorem B such that m(x) > o be a Boolean algebra. By the Stone Representation i s isomorphic to a f i e l d of subsets A measure on m(x) >_ o B B i s any function for x e B m : B •> R of some set X . such that and m ( x v y ) = m(x) + m(y) X A y = o . A measure for x ^ o . m : B -* R F m(o) = o and f o r x, y e.B i s s t r i c t l y positive i f 12 Theorem (Tarski) Any measure m defined on a subalgebra Q may be extended to a measure m on B B^ of a Boolean algebra such that the range of m B lies w i t h i n the closure of the range of m Proof By the Stone Representation B^CZ B are f i e l d s of subsets of some set X . Let any f i n i t e , d i s j o i n t subset a Q(£,n) mean that 1 > m =- U ' n K II r e f i n e s be c a l l e d £ . Thus i f f o r j = l,...,m . E v i d e n t l y Q then i s concurrent (since 3 J Thus there e x i s t s some f o r a l l B -measurable p a r t i t i o n s Q i s an i n t e r n a l , , Write n Q = {n 1 n ^-finite, co o n '"A = Un R ir c A R A eB \ IT such that Q of X . In p a r t i c u l a r , B ~measurable p a r t i t i o n of II } . Consequently (l) for X =Un and n = {n ,...,n }. and Q(£,n) i s a f i e l d of subsets). Q(£,IIQ) 11^ of nonvoid sets such that o I = {I --.,I } o HC B o B ~measurable p a r t i t i o n of X . For such p a r t i t i o n s l e t the binary relation B theorem i t may be assumed that J X . 13 • Let A {x..,...,x } be an i n t e r n a l , * - f i n i t e sequence i n X such 1 0) that x E IT f o r K = l,...,w . Thus (1) may be w r i t t e n as K. K (2) *A = U * V for A £ B o . K Therefore (3) for n A m (A) = *m (*A) = o o A £ B . Extend o m on o B o £ *m (II ) '-^ o K £ A to a A measure m. 1 on B by letting (4) I m.(A) = J- X E K for m o A E B • Note that for *m (n ) m m(A) = o A m^(A) .on B • to a measure o (5) jA. K i s f i n i t e f o r A e B . Now, extend on B by l e t t i n g ' \(A) A £ B. . F i n a l l y , l e t A e B . Then m^A) - m(A) but A m^(A) e 2.2 range (m) . Hence m(A) l i e s w i t h i n the closure of range (m) Nikodym's Result Theorem (Nikodym) Let F B be a Boolean algebra. and a measure m : B -> F Then there e x i s t s an ordered f i e l d which i s s t r i c t l y p o s i t i v e . 14 Proof By the Stone Representation theorem i t may be assumed that B i s a f i e l d of subsets of some set X . Let IT = {II, ,...,11 } be Q 1 to * ft an i n t e r n a l , * - f i n i t e , B-measurable p a r t i t i o n of X such that i f A e B then ft A = U n ^ K C IT. Let m(A) = £ m ( \ ) where m(II ) i =- nr*A K A ft for K = l,...,co . Evidently m : B -> R ft measure. But R i s an ordered f i e l d . 2.3 i s a strictly positive Boolean Covers definition Let and l e t x X be a set and l e t F be a function from F be a f i e l d of subsets of into S(X) . Then x X i s called pre-Boolean i f (1) x(X) = X (2) T(4>) = <f> (3) x(A H B) = x(A) 0 x(B) • . Furthermore, i f (4) then F x into x(A U B) = x ( A ) U i s c a l l e d Boolean. S(X) such that T(B) A cover f o r x x ( A ) C a (A) i s any f u n c t i o n for A e F . a from 15 Lemma Let T : F -> S(X) be pre-Boolean. some A x * F e such that Then f o r each x(A) = {x e X : A x C * A} x e X there e x i s t s for A e F . Proof For each x e X l e t F = {A e F : x e x(A)} . Then F x x has the f i n i t e i n t e r s e c t i o n property so the binary r e l a t i o n QCA^jA^) F^ meaning 2^~ "*" * A„ e F such that Q(A,A„) Q x Q on for that A sc o n c u r r e n t - Thus there e x i s t s some for A e F x . In p a r t i c u l a r , A e F . Let A = A„ . Now, l e t A e F . x x Q x(A) = (x E X : A C x A A} . To show that A C Q n * A I t s u f f i c e s to show x ( A ) C {x e X : A d x * A} * let x e x(A) . Then A e F so A C A . To show that x x * * ' x(A)Z) {x e X : A C A} l e t x e X and A CZ A . Then the sentence x x 3 B(BC A i s true i n and * X and x e x(B)) ' ( l e t B = A ) so i t i s true i n X . But x x(B) = x(B O A) = x(B) C\ x(A) so x e x(A) . x e x(B) Theorem Let x : F -> S(X) be pre-Boolean. Then x has a Boolean cover. Proof For each for x e X s e l e c t some A A e F . A l s o , f o r each x x e X e F such that A C x x x s e l e c t some y e A . Let x x A 16 a(A) = {x e X : y e Boolean. i s a Boolean cover f o r 2.4 Thus a A} . Then a T . covers Evidently a is T . Remarks In 1962, W. A. J . Luxemburg [Luxemburg 1962a, 1962b] employed ultrapowers to prove both Tarski's r e s u l t [Tarski 1930] and Nikodym's r e s u l t [Nikodym 1956, 1960] concerning extensions of measures on Boolean algebras. In both proofs, however, the existence of a measure about a s i n g l e point i n a Boolean algebra was required before the methods of ultrapowers could be applied. For Any measure example, Tarski's r e s u l t depends upon the f o l l o w i n g . m defined on a subalgebra Q may be extended to a measure m^ on B^ B^ and some point moreover, requires the f o l l o w i n g . b ^ x e B X q q of a Boolean algebra such that the range of l i e s w i t h i n the closure of the range of generated by B m (where Q B^ B m^ i s the subalgebra ) . The r e s u l t of Nikodym, If then there e x i s t s a measure B i s a Boolean algebra and m on B such that m(x) ^ o . In t h i s thesis we have used Robinson's enlargements [Robinson 1974] to remove such requirements by providing the desired extensions immediately. The proof of the r e s u l t concerning Boolean covers i s an improvement of a note i n the theory of l i f t i n g due to E i f r i g 1972], [Eifrig Though E i f r i g employs enlargements h i s method i s unnecessarily complicated. Some remarks about Boolean covers and l i f t i n g serve to demonstrate t h i s . Let < f i , F, P> be a complete p r o b a b i l i t y space 17 (where P i s the countably a d d i t i v e measure on the Borel f i e l d and l e t N = {A e F : P(A) = 0 } . I t i s easy to show subsets of ft ) that the r e l a t i o n relation. F of A ~ B meaning A A B e N i s an equivalence A density on <ft, F, P^> i s any f u n c t i o n 9 : F ->-. F satisfying (1) 0(A) ~ A (2) A - B (3) 8(<f>) = <j> and (4) 0(A O B) = 9 ( A ) O 9(B) . implies 9(A) = 9(B) 9 (fl) = fl Furthermore, i f 0 s a t i s f i e s (5) O(AL)B) = 8(A) U 9(B) then i t i s c a l l e d a l i f t i n g on <[fl, F, P^> . E i f r i g provides a non-standard proof of the well-known r e s u l t that any density on <Cfl, F, ?y may be extended to a l i f t i n g on <Cft, F, P/> by employing the f a c t that f o r sets of measure 1, i n c l u s i o n i s a concurrent binary relation. But such considerations may be avoided by noting that any density i s pre-Boolean and hence possesses a Boolean cover. Furthermore, i t i s easy to show that any Boolean cover of a density ( i n a complete p r o b a b i l i t y space) i s already a l i f t i n g . 18 BIBLIOGRAPHY Bachman, G . / N a r i c i , L. 1966 Functional A n a l y s i s . Academic Press. New York. E i f r i g , B. 1972 E i n Nicht-Standard-Beweis f u r d i e Existenz eines L i f t i n g s . Arch. Math. 23, 425-427. Kelley, J. 1955 General Topology. Van Nostrand. New York. Luxemburg, W. A. J ; 1962a Non-Standard A n a l y s i s . Lectures on A. Robinson's theory of i n f i n i t e s i m a l s and i n f i n i t e l y large numbers. Pasadena. 1962b Two a p p l i c a t i o n s of the method of construction by ultrapowers to A n a l y s i s . B u l l e t i n of the American Mathematical Society ser. 2, 68, 416-419. Nikodym, 0. 1956 On extension of a given f i n i t e l y a d d i t i v e , f i e l d valued, non-negative measure, on a f i n i t e l y a d d i t i v e Boolean t r i b e , to another t r i b e more ample. Rend. Sem. Mat. Univ. Padova. 26, 232-327. 1960 Sur l a mesure non-archimedienne e f f e c t i v e sur une t r i b e de Boole a r b i t r a i r e . C. R. Acad. S c i . P a r i s 251, 2113-2115. Robinson, A. 1974 Non-Standard A n a l y s i s . North-Holland P u b l i s h i n g Co. Amsterdam. T a r s k i , A. 1930 Une c o n t r i b u t i o n a l a thdorie de l a mesure. 42-50. Fund. Math. 15,
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Non-standard analysis Cooper, Glen Russell 1975
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Title | Non-standard analysis |
Creator |
Cooper, Glen Russell |
Date Issued | 1975 |
Description | In this thesis some classical theorems of analysis are provided with non-standard proofs. In Chapter 1 some compactness theorems are examined. In 1.1 the monad μ(p) of any point p contained in a set X (and relative to a family H of subsets of X ) is defined. Using monads, a nonstandard characterization of compact families of subsets of X is given. In 1.2 it is shown that the monad μ(p) of any point p ε X (relative to H ) remains unchanged if H is extended to the smallest topology τ(H) on X containing H . Then, as an immediate consequence, the Alexander Subbase theorem is proved. In 1.3 monads are examined in topological products of topological spaces. Then, in 1.4 and 1.5 respectively, both Tychonoff's theorem and Alaoglu's theorem are easily proved. In Chapter 2 various extension results of Tarski and Nikodým (in the theory of Boolean algebras) are presented with rather short proofs. Also, a result about Boolean covers is proved. The techniques of non-standard analysis contained in Abraham Robinson's book (Robinson 1974) are used throughout. The remarks at the end of each chapter set forth pertinent references. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080095 |
URI | http://hdl.handle.net/2429/19245 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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