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UBC Theses and Dissertations

Non-standard analysis Cooper, Glen Russell 1975

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NON-STANDARD ANALYSIS by GLEN RUSSELL COOPER B.Sc, University 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 this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1975 In 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 m a k e 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 a n d 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 H e a d 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 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 V a n c o u v e r 8, C a n a d a i i Supervisor: Dr. P. Greenwood ABSTRACT In t h i s thesis some c l a s s i c a l theorems of analysis are provided with non-standard proofs. In Chapter 1 some compactness theorems are examined.. In 1.1 the monad u(p) of any point p contained i n a set X (and r e l a t i v e to a family H of subsets of X ) i s defined. Using monads, a non-standard characterization of compact families of subsets of X i s given. In 1.2 i t i s shown that the monad u(p) of any point p e X (r e l a t i v e to H ) remains unchanged i f H i s extended to the smallest topology T(H) on X containing H . Then, as an immediate consequence, the Alexander Subbase theorem i s proved. In 1.3 monads are examined i n topological products of topological spaces. Then, i n 1.4 and 1.5 respectively, both Tychonoff's theorem and Alaoglu's theorem are ea s i l y proved. In Chapter 2 various extension results of Tarski and Nikodym (in the theory of Boolean algebras) are presented with rather short proofs. Also, a result about Boolean covers i s proved. The techniques of non-standard analysis contained i n Abraham Robinson's book [Robinson 1974] are used throughout. The remarks at the end of each chapter set forth pertinent references. i i i TABLE OF CONTENTS . Page Introduction 1 Chapter 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 Chapter 2 2.1 Tarski's Result 11 2.2 Nikodym's Result 13 2.3 Boolean Covers 14 2.4 Remarks 16 Bibliography 18 i v ACKNOWLEDGEMENT S I wish to thank my advisor, P r i s c i l l a Greenwood, for proposing this thesis and guiding i t s development. As w e l l , I wish to thank Andrew Adler for his kind c r i t i c i s m . F i n a l l y , I wish to thank the Department for providing f i n a n c i a l assistance. This thesis i s dedicated to my wife, Adriana (nee Buonanno) Cooper. i INTRODUCTION Some techniques of non-standard analysis contained i n Abraham Robinson's book [Robinson 1974] are reviewed here. Recall that i f ^ X T^ i s t n e f u l l , normal higher order . structure on a set X then there exists a normal higher order structure { X } on a set X such that { X } extends {X } (i n p a r t i c u l a r T T X X contains X ) and such that the following properties are s a t i s f i e d . I f S i s a higher order sentence i n {X } and S i s i t s T r* * interpretation i n { X } then S i s true i n {X } i f f S • i s true X X i n { X } . This property i s sometimes cal l e d the transfer p r i n c i p l e . For the next property we r e c a l l some d e f i n i t i o n s . Suppose 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 i n {X } such that Q(x,y) for some y in {X^ _} . The range of Q , range (Q) , i s defined s i m i l a r l y . A binary r e l a t i o n Q(x,y) i n {X } i s call e d concurrent i f whenever , x^,...,x e domain (Q) there exists some y e range (Q) such that Q(x^,y) for K = l, . . . , n . I f Q(x,y) i s a concurrent, binary r e l a t i o n i n {X } then there exists some y„ i n { X } (called an . x •'Q x . * i d e a l element for Q ) such that Q(x,yp) i s true i n { X^ _} for x e domain (Q An enlargement of the f u l l , normal higher order structure {X^ .} on a set X i s any normal higher order structure { X^ _} on a set X sa t i s f y i n g the above properties. Thus, by the above, any f u l l , normal higher order structure possesses and enlargement (which therefore s a t i s f i e s the transfer 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 structure {. j on a set X i s an enlargement of the f u l l , normal higher order structure {X^ _} on a set X . Recall that, i n general, { X^} need not be f u l l . AA A Thus i f { 1 S the f u l l , normal higher order structure on X AA A then, for each x , X^ - X^ need not be empty. Any r e l a t i o n AA A contained i n some X - X i s called external while the relations T T A contained i n X are called 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 called standard. T Some general properties of enlargements are recalled here. A ft Suppose the normal higher order structure { X^} on the set X i s an enlargement for the f u l l , normal higher order structure {X^} on the set X . Let A and B be sets (that i s , unary relations) i n {X •}• . Then A U B i s contained i n {X } since {X } i s f u l l . Let T X X A A A A A, B and (A U B) be the extensions i n { X } of A, B and , x AL) B respectively. Then * ( A U B) = *A U *B . Similar results hold for a l l f i n i t e Boolean combinations of sets. A A Let { R } be a normal higher order structure on a set R x . ° which i s an enlargement of the f u l l , normal higher order structure {R } on the set R of r e a l numbers. Recall that, by transfer, the T 1 sentence i n {R } which states that R i s an ordered f i e l d under the x A A usual relations also states (when interpreted i n { R T J ) that R i s A an ordered f i e l d under the same relations (when extended i n { R }) . x A A Thus R ( s t r i c t l y speaking { R^} ) may be called a f i e l d of non-standard r e a l numbers. Let be the absolute value function i n { R } . i i T A Recall that a non-standard r e a l number x e R i s called f i n i t e i f 3 A . . |x| <_ y for some y e R (otherwise i t i s cal l e d i n f i n i t e ) . Also, A a non-standard r e a l number x e R i s called i n f i n i t e s i m a l i f |x| < y for a l l positive y e R . Clearly o e R i s i n f i n i t e s i m a l . But the binary r e l a t i o n Q(x,y) i n meaning o < y < x i s A concurrent. Thus there exists some y^ i n { R^ _} such that Q(x,y^) for a l l positive x e R . Hence 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 th i s i t i s easy to show that R A i s non-archimedean. Recall that i f x e R i s a f i n i t e non-standard r e a l number then there exists a unique standard r e a l number °x e R (called the standard part of x ) such that x - °x i s i n f i n i t e s i m a l (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 x be defined as the set of a l l f i n i t e y e R sa t i s f y i n g y - x . Recall that a set A C 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 is.closed 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 of standard r e a l numbers i s f i n i t e i f f there exists a bisection from {1,...,n} onto A A (for some n e N ) . An inte r n a l set A C R of non-standard r e a l numbers i s called ^ - f i n i t e i f f there exists an in t e r n a l bisection A A from {1,...,(JJ} onto A (for some co e N ) . A si m i l a r d e f i n i t i o n exists for ^ - f i n i t e i n t e r n a l sequences. 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 structure {X } on a set X contains the set R of standard T r e a l numbers (extend X to X U R i f necessary) and thus any enlargement of {X_^ } contains a set R of non-standard r e a l numbers. From now on such enlargements are assumed fixed i n any discussion. CHAPTER 1 1.1 Compact Families of Sets  d e f i n i t i o n Let X be a set and l e t S(X) be the power set of X and l e t H CS(X) be a set of subsets of X . If x e X then ft y(x) = Pi { A : x e A and A e H} i s called the monad of x ft ( r e l a t i v e to H ) . A point p e X i s called near-standard (r e l a t i v e to H ) i f p e u(x) for some x e X (write p - x ( r e l a t i v e to H )) . A set X i s called H-compact i f every cover of X by members of H has a f i n i t e sub-cover. Note that i f H i s a topology on X then H-compactness i s just 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 for each x e X there exists i * some A e H such that x e A but p 4 A However x x T x {A : x e X} covers X so there exists some f i n i t e set • x {•x, ,. . . ,x } C X such that X = A U ' " L J A . Thus I n x.. x 1 n X '= (A U "•L)A ) = A U • ' "U A so p e A for some X-, . X X-, X X T 7. ' 1 n 1 n K K = 1 , . . .,n . This contradiction implies that p i s near-standard ( r e l a t i v e to H ) . Thus each p e X i s near-standard ( r e l a t i v e to H ) . Suppose, conversely, each p e X i s near-standard ( r e l a t i v e to H ) but assume X i s not H-compact. Then there exists some {A } C H covering X 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 exists some x^ e X such that x^ 4 A for a l l a . However Q Q T a x^ - x (r e l a t i v e to H ) for some x e X . But x e A for some a . Q a * Thus x„ e A This contradiction implies that X i s H-compact. Q a 1.2 Alexander Subbase Theorem d e f i n i t i o n > • 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 X containing H . Recall that T ( H ) = H ' U {X} where H ' i s the set of a l l unions of f i n i t e intersections 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 for some f i n i t e set {A,,...,A } C H • 1 n I n 6 Theorem Let X be a set and l e t H be a set of subsets of X and l e t x(H) be the smallest topology on X containing H . Furthermore, l e t p e X and x e X . Then p - x (r e l a t i v e to H ) i f f p - x (re l a t i v e to x(H) ) . Proof Suppose p - x (re l a t i v e to H ) . I t suffices to show that p - x (r e l a t i v e to x(H) ) . Suppose x e A and A e x(H) . I t ft * suffices to show that p e A . If A = X then p e A . Thus, assume A ^ X . Then A e h" where H' i s the set of a l l unions of f i n i t e intersections of members of H . Thus x e A,H '"OA C A 1 n for some f i n i t e set {A^ A^}C H . In p a r t i c u l a r , x e A^ ft for • K = 1,...,n . Hence p e A^ for K = 1,...,n since p - x (re l a t i v e to H ) . But then p e *A. f] ' ' ' 0 *A = * (An 0 • • • 0 A ) C *A 1 n 1 n ft so p e A as desired. The reverse implication i s clear. Corollary (Alexander Subbase) Let X be a set and l e t H be a set of subsets of X and l e t x(H) be the smallest topology on X containing H . If X i s H-compact then X i s x(H)-compact. Proof ft Suppose X i s H-compact. Let p e X . By 1.1 there exists some x e X such that p - x ( r e l a t i v e to H ) . By 1.1 i t suffices to show that p - x (r e l a t i v e to T(H) ) . But this follows from the above theorem. 1.3 Products of Topological Spaces  d e f i n i t i o n If {(X^,T_^) : i e 1} i s a family of topological spaces (where x_^  i s the topology on X^) l e t (ITX^, IIT_) be i t s topological product (where IIx i s the product topology on IIX ) . Recall that the product topology on IIX^ i s the smallest topology on 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. e x . . Hence the product topology on IIX. i s X l 1 1 1 the set of a l l unions of sets of the form n A = O {f e nx. : f ( i ) e A. } K=l 1 k XK where i . . , . . . , i e l and A. e x . for K = l , . . . , n . 1 n XK XK Lemma Let {(X_^,x^) : i e 1} be a family of topological spaces and l e t ft ( n x ^ n x j be i t s topological product. Suppose g e (rOL) and h e IIXi . Then g - h (r e l a t i v e to Tlx ) i f f g(i) - h(i) ( r e l a t i v e to • x ) for i e l Proof Suppose g - h (r e l a t i v e to ) . Let i ^ e I . I t suffices to show that g(i-,) --h(i-) ( r e l a t i v e to x. ) . Let * h ( i ) e A. and A. e x . . I t suffices to show that g ( i ) e A. 1 1 Xl Xl X l But h e {f e nx. : f ( i j e A. } and {f e nx. : f ( i , )• e A. } e nx. x 1 x^ x 1 x^ x Hence g e {f e nx_^ : f ( i ^ ) e A_^  } since g - h (r e l a t i v e to nx_^  ft 1 Thus g(i.),e A. as desired. 1 1 Suppose, conversely, g(i) - h(i) ( r e l a t i v e to x^ ) for l e i . I t suffices to show that g - h (r e l a t i v e to nx^ ) . Let h e A and A e nx. . Then i t suffices to show that x * ' £ g e .A . But h e r J {f e HX. : f(i„) e A. }CA for some K—X X K i ' K i ,...,i e I and some A. e x . for K = l , . . . , n . In p a r t i c u l a r h ( i ) e A. and A. e x . for K = l,. . . , n . But g(i„) - h(i 7 r) iv x^ . x R x R K K (re l a t i v e to x. ) for K = l, . . . , n so g ( i ) e A. for XK K V ft K = l , . . . , n . Hence g e {f e JJX. : f ( i ) e A. } for K = l, . . . , n 1 K XK n A Therefore, g e H '{f e nx. : f ( i T J e A. } = K—X X K Xj^ 1 1 ft ft * 0 {f e nx. : f ( i ) e A. } C A . Thus g e A as desired, x x K 1 K 1.4 Tychonoff's Theorem  Theorem (Tychonoff) Let { ( X ^ x ^ ) : i e 1} be a family of topological spaces and l e t (ITX^IIT^) be i t s topological product. Suppose X_^  i s x^-compact for i e I . Then IT.X. i s Tlx . -compact. x x Proof' . ' A. Suppose X^ i s x^-compact for i £ I . Let g E (IIX^) . By 1.1 i t suffices to show that g i s near-standard ( r e l a t i v e to A II ) . But g(i) e for i e I . By 1.1, g(i) i s near-standard (r e l a t i v e to x. ) since X. i s x.-compact, x x x Thus g(i) - x. (re l a t i v e to x. ) for some x. e X. . " 1 x x x Let h(i) = x ± for i E I . Then h E -IIX and g(i) - h ( i ) (re l a t i v e to x_^ ) for i e I . By 1.3, g - h (r e l a t i v e to JIx^) . Thus g i s near-standard (r e l a t i v e to IIx^ ) as desired. 1-5 Alaoglu's Theorem  d e f i n i t i o n 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, li n e a r functionals on X . Recall that x e X and f e L(X) implies | | f ( x ) | | <_ | | f | | | |x| | (where || || denotes the norms on X and L(X) ) . Note that L(X) i s contained i n the X Cartesian power R . Let x^ be the topology of pointwise convergence X * y X on R . By 1.3, i f g E (K ) and h e R then g - h ( r e l a t i v e to x^ ) i f f g(x) - h(x) for X E X . Let B = {f E L(X) : ||f||£-l}. 10 Theorem (Alaoglu) B i s x -compact P Proof Let g e B . By 1.1 i t suffices to show that g - h * ( r e l a t i v e to x ) for some h e B . I f x E X then P ,*llg(x)II 1 *l|g|1*1 M | £ *| M | (since g e * ( L ( X ) ) and * II g I I — x ) • • 1 1 1 p a r t i c u l a r , i f x £ X then | | x | | = | | x | | so I|g(x)I I 1 I l xlI • Hence g(x) i s f i n i t e and °g(x) i s defined i f x E X . Let h(x) = °g(x) for x E X . Then h E B . But g(x) - h(x) for x E X . Thus, by 1.3, g - h ( r e l a t i v e to x^ ) . 1.7 Remarks In 1.1 a generalization i s provided for Robinson's non-standard characterization of compactness i n topological spaces [Robinson 1974]. This i s done by observing that the d e f i n i t i o n of compactness, as we l l as Robinson's characterization of i t , does not require the axioms for a topological space. Then, i n 1.2, this generalization yields the Alexander subbase theorem. In 1.3 the near-standard r e l a t i o n i n topological products i s reduced to one involving the coordinate spaces. Then, i n 1.4, Tychonoff's theorem i s e a s i l y proved. Both 1.3 and 1.4 appear i n a book of Robinson [Robinson 1974]. The usefulness of 1.3 may be seen elsewhere, however. For instance, Alaoglu's theorem i s proved i n 1.5. This i s possible 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 results 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 yields Alaoglu's theorem by providing a compact topological product i n which the unit b a l l of the dual of a normed li n e a r space exists as a closed (hence compact) subspace [Bachman-Narici 1966]. Thus i n the standard proof of Alaoglu's theorem some analysis i s required to show that the above subspace i s closed (hence compact). The lemma i n 1.3 eliminates this requirement by providing compactness d i r e c t l y . CHAPTER 2 2.1 Tarski's Result  d e f i n i t i o n Let B be a Boolean algebra. By the Stone Representation theorem B i s isomorphic to a f i e l d of subsets F of some set X . A measure on B i s any function m : B -* R such that m(o) = o and m(x) >_ o for x e B and m(xvy) = m(x) + m(y) for x, y e.B such that X A y = o . A measure m : B •> R i s s t r i c t l y positive i f m(x) > o for x ^ o . 12 Theorem (Tarski) Any measure mQ defined on a subalgebra B^ of a Boolean algebra B may be extended to a measure m on B such that the range of m l i e s within the closure of the range of m Proof By the Stone Representation theorem i t may be assumed that 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 H C B of nonvoid sets such that X =Un be called o a Bo~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 r e l a t i o n Q(£,n) mean that II refines £ . Thus i f I = { I 1 > - - . , I m } and n = {n ,...,n }. and Q(£,n) then =- U ' n for j = l,...,m . Evidently Q i s concurrent (since K 3 B i s a f i e l d of subsets). Thus there exists some IT such that o J Q Q(£,IIQ) for a l l BQ-measurable pa r t i t i o n s \ of X . In p a r t i c u l a r , 11^ i s an internal,, ^ - f i n i t e , B o~measurable p a r t i t i o n of X . Write n = {n n II } . Consequently Q 1 co n J ( l ) ' " A = U n R i r R c A for A e B 13 • A Let {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 for K = l,...,w . Thus (1) may be written as K. K (2) *A = U * n K V A for A £ B . Therefore o (3) m (A) = *m (*A) = £ *m (II ) o o '-^  o K £ A A for A £ B . Extend m on B to a measure m. on B by o o o 1 l e t t i n g (4) m.(A) = I *m (n ) J- j . o K A X K E A for A E B • Note that m^(A) i s f i n i t e for A e B . Now, extend m .on B • to a measure m on B by l e t t i n g o o ' (5) m(A) = \(A) for 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 range (m) . Hence m(A) l i e s within the closure of range (m) 2.2 Nikodym's Result  Theorem (Nikodym) Let B be a Boolean algebra. Then there exists an ordered f i e l d F and a measure m : B -> F which i s s t r i c t l y p o s itive. 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 ft i then A = U ^ IT. Let m(A) = £ m ( \ ) where m(II ) = -n K C A n r * A K ft for K = l,...,co . Evidently m : B -> R i s a s t r i c t l y positive ft measure. But R i s an ordered f i e l d . 2.3 Boolean Covers  d e f i n i t i o n Let X be a set and l e t F be a f i e l d of subsets of X and l e t x be a function from F into S(X) . Then x i s cal l e d 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) x(A U B) = x(A)U T ( B ) then x i s cal l e d Boolean. A cover for x i s any function a from F into S(X) such that x ( A ) C a (A) for A e F . 15 Lemma Let T : F -> S(X) be pre-Boolean. Then for each x e X there exists * * some A e F such that x(A) = {x e X : A C A} for A e F . x x 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 intersection property so the binary r e l a t i o n QCA^jA^) on F^ meaning A2^~ "*"s c o n c u r r e n t - Thus there exists some * * A„ e F such that Q(A,A„) for A e F . In p a r t i c u l a r , An C A Q x Q x Q for A e F . Let A = A„ . Now, l e t A e F . I t suffices to show x x Q A * that x(A) = (x E X : A C A} . To show that x ( A ) C {x e X : A d A} x x * l e t 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 and x e x(B)) * ' i s true i n X (let B = A ) so i t i s true i n X . But x e x(B) x and x(B) = x(B O A) = x(B) C\ x(A) so x e x(A) . Theorem Let x : F -> S(X) be pre-Boolean. Then x has a Boolean cover. Proof For each x e X select some A e F such that A C A x x x for A e F . Also, for each x e X select some y e A . Let x x x 16 a(A) = {x e X : y e A} . Then a covers T . Evidently a i s Boolean. Thus a i s a Boolean cover for T . 2.4 Remarks In 1962, W. A. J . Luxemburg [Luxemburg 1962a, 1962b] employed ultrapowers to prove both Tarski's result [Tarski 1930] and Nikodym's result [Nikodym 1956, 1960] concerning extensions of measures on Boolean algebras. In both proofs, however, the existence of a measure about a single point i n a Boolean algebra was required before the methods of ultrapowers could be applied. For example, Tarski's result depends upon the following. Any measure mQ defined on a subalgebra B q of a Boolean algebra B may be extended to a measure m^  on B^ such that the range of m^  l i e s within the closure of the range of mQ (where B^ i s the subalgebra generated by B^ and some point X q ) . The result of Nikodym, moreover, requires the following. I f B i s a Boolean algebra and b ^ x e B then there exists a measure m on B such that m(x) ^ o . In this thesis we have used Robinson's enlargements [Robinson 1974] to remove such requirements by providing the desired extensions immediately. The proof of the result 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 [ E i f r i g 1972], Though E i f r i g employs enlargements his 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 <fi, F, P> be a complete probability space 17 (where P i s the countably additive measure on the Borel f i e l d F of subsets of ft ) and l e t N = {A e F : P(A) = 0 } . I t i s easy to show that the r e l a t i o n A ~ B meaning A A B e N i s an equivalence r e l a t i o n . A density on <ft, F, P^> i s any function 9 : F ->-. F sa t i s f y i n g (1) 0(A) ~ A (2) A - B implies 9(A) = 9(B) (3) 8(<f>) = <j> and 9 (fl) = fl (4) 0(A O B) = 9(A)O 9(B) . 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 cal 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 result 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 fact that for sets of measure 1, inclusion i s a concurrent binary r e l a t i o n . 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 (in a complete probability space) i s already a l i f t i n g . 18 BIBLIOGRAPHY Bachman, G./Narici, L. 1966 Functional Analysis. Academic Press. New York. E i f r i g , B. 1972 Ein Nicht-Standard-Beweis fur die 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 Analysis. Lectures on A. Robinson's theory of in 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 applications of the method of construction by ultrapowers to Analysis. 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 additive, f i e l d valued, non-negative measure, on a f i n i t e l y additive 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 effec 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 . Paris 251, 2113-2115. Robinson, A. 1974 Non-Standard Analysis. North-Holland Publishing Co. Amsterdam. Tarski, A. 1930 Une contribution a l a thdorie de l a mesure. Fund. Math. 15, 42-50. 

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