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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Non-standard analysis Cooper, Glen Russell 1975
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Title | Non-standard analysis |
Creator |
Cooper, Glen Russell |
Publisher | University of British Columbia |
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 |
AggregatedSourceRepository | DSpace |
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