The Theory of Measures and Integration

Transcription

1 The Theory of Measures ad Itegratio A Solio Maual for Vestrup (2003) Jiafei She School of Ecoomics, The Uiversity of New Soh Wales Sydey, Australia

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3 I hear, I forget; I see, I remember; I do, I uderstad. Old Proverb

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5 Cotets Preface Ackowledgemets ix xi Set Systems Systems, -Systems, ad Semirigs Systems System Semirig Fields Fields The Borel -Field Measures Measures Cotiuity of Measures A Class of Measures Extesios of Measures Extesios ad Restrictios Oer Measures Carathéodory s Criterio Existece of Extesios Uiqueess of Measures ad Extesios The Completio Theorem The Relatioship betwee.a/ ad M. / Approximatios A Further Descriptio of M. / Lebesgue Measure Lebesgue Measure: Existece ad Uiqueess Lebesgue Sets Traslatio Ivariace of Lebesgue Measure v

6 vi CONTENTS 5 Measurable Fuctios Measurability Combiig Measurable Fuctios Sequeces of Measurable Fuctios Almost Everywhere Simple Fuctios Some Covergece Cocepts Cotiuity ad Measurability A Geeralized Defiitio of Measurability The Lebesgue Itegral Stage Oe: Simple Fuctios Stage Two: Noegative Fuctios Stage Three: Geeral Measurable Fuctios Stage Four: Almost Everywhere Defied Fuctios Itegrals Relative to Lebesgue Measure Semicotiuity The L p Spaces L p Space: The Case 6 p < C The Riesz-Fischer Theorem L p Space: The Case 0 < p < L p Space: The Case p D C Cotaimet Relatios for L p Spaces Approximatio More Covergece Cocepts Prelude to the Riesz Represetatio Theorem The Rado-Nikodym Theorem The Rado-Nikodym Theorem, Part I Products of Two Measure Spaces Product Measures The Fubii Theorems Arbitrary Products of Measure Spaces Notatio ad Covetios Costructio of the Product Measure Refereces

7 List of Figures. Iclusio betwee classes of sets A A [ B 2 M./ f ad f f! f f p ad f q L f! p 0, b f! vii

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9 Preface Sydey, Jiafei She ix

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11 Ackowledgemets xi

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13 SET SYSTEMS Remarks Remark.. Kleke (2008, Fig.., p.7) provides a chart to idicate the relatioships amog the set systems. Here I replicate his chart; see Figure.. -field Field 2 A -[-stable 2 A \-stable X-closed [-closed -rig -system 2 A A B H) B X A 2 A F A 2 A 2 A Rig 2 A X-closed [-closed -[-stable Semirig [-stable 2 A B X A D F id B i \-closed Figure.. Iclusio betwee classes of sets A 2 Semirig S -stable! Rig See part (g) of Exercise.22;

14 2 CHAPTER SET SYSTEMS -rig 2A! -field See part (b) of Exercise.43; Rig 2A! Field 2 A ad A is closed uder differece implies that A 2 A H) A c D A 2 A; -systme T -stable! -field See Exercise.0. Remark.2. This otes is for Exercise.34 (p.7). See Kleke (2008, Example.40, p.8-9). We costruct a measure for a ifiitely ofte repeated radom experimet with fiitely may possible ocomes (Product measure, Beroulli measure). Let S be the set of possible ocomes. For s 2 S, let p s > 0 be the probability that s occurs. Hece P s2s p s D. For a fixed realizatio of the repeated experimet, let z.!/; z 2.!/; : : : 2 S be the observed ocomes. Hece the space of all possible ocomes of the repeated experimet is D S N. We defie the set of all sequeces whose first values are z.!/; : : : ; z.!/: Œz.!/; : : : ; z.!/ D! 0 2 W z i.! 0 / D z i.!/ for ay i D ; : : : ; : (.) Let C 0 D f g. For 2 N, defie the class of cylider sets that deped oly o the first coordiates C D fœz.!/; : : : ; z.!/ W z.!/; : : : ; z.!/ 2 Sg ; (.2) ad let C S D0 C. We iterpret Œz.!/; : : : ; z.!/ as the evet where the ocome of the first experimet is z.!/, the ocome of the secod experimet is z 2.!/ ad fially the ocome of the -th experimet is z.!/. The ocomes of the other experimets do ot play a role for the occurrece of this evet. As the idividual experimets ought to be idepedet, we should have for ay choice z.!/; : : : ; z.!/ 2 E that the probability of the evet Œz.!/; : : : ; z.!/ is the product of the probabilities of the idividual evets.. -Systems, -Systems, ad Semirigs.. -Systems I Exercise.3 (..). Let D. ; ˇ. Let P cosists of alog with the rsc subitervals of. P is a -system of subsets of. ; ˇ. Proof. Let A D.a; b ad B D.c; d be P -sets. The either A \ B D 2 P, or A \ B D.a _ c; b ^ d 2 P. I Exercise.4 (..2). Must be i every -system? Solio. Not ecessary. For example, let

15 SECTION. -SYSTEMS, -SYSTEMS, AND SEMIRINGS 3 D.0; ; A D.0; =2 ; B D.=4; ; C D.=4; =2 ; ad let P D fa; B; C g. The P is a -system o, ad P. Geerally, if A \ B for ay A; B i a -system, the does ot i this -system. I Exercise.5 (..3). List all -systems cosistig of at least two subsets of f! ;! 2 ;! 3 g. Solio. These -systems are: f! i g; f! i ;! j g,.i; j / 2 f; 2; 3g 2 ad j i; f! i g; f! ;! 2 ;! 3 g ; f! i ;! j g; f! ;! 2 ;! 3 g ; f! i g; f! i ;! j g; f! ;! 2 ;! 3 g ; ; f! i g; f! i ;! j g, i D ; 2; 3, ad j i; ; f! i ;! j g; f! ;! 2 ;! 3 g ; ; f! i g; f! i ;! j g; f! ;! 2 ;! 3 g. I Exercise.6 (..4). If P k cosists of the empty set ad the k-dimesioal rectagles of ay oe form, the P k is a -system of subsets of R k. Proof. Let A; B 2 P k be two k-dimesioal rectagles of ay form. We also write A D A A 2 A k ad B D B B k, where A i ad B i are rsc itervals for every i 2 f; : : : ; g. We also assume that A ad B ; for otherwise A \ B D 2 P k is trivial. The A \ B D.A A k / \.B B k / D k id.a i \ B i / 2 P k sice A i \ B i is a rsc iterval i R. I Exercise.7 (..5). Let P cosist of ad all subsets of R k that are either ope or closed. The P is ot a -system of subsets of R k. Proof. To get some ituitio, let k D. Cosider two P -sets: A D.0; =2 ad B D Œ=4; /. Note that either A or B are ope or closed o R, b their itersectio A \ B D Œ=4; =2 is closed o R, ad is ot i P. Now cosider the k-dimesioal case. Let A; B 2 P ; let A D k id A i ad B D k id B i ; particularly, we let A i D.a i ; b i ad B i D Œc i ; d i /, where a i < c i < b i < d i. The.a i ; b i \ Œc i ; d i / D Œc i ; b i, ad A \ B D k id.a i \ B i / D k id Œc i; b i is closed o R k. I Exercise.8 (..6). For each i a oempty idex set A, let P be a - system over.

16 4 CHAPTER SET SYSTEMS a. The collectio T 2A P is a -system o. b. Let A 2. Suppose that fp W 2 Ag is the exhaustive list of all the - system that cotai A. I other words, each P A, ad ay -system that cotais A coicides with some P. The T 2A P is a -system that cotais A. If Q is a -system cotaiig A, the T 2A P Q. The miimal -system geerated by A always exists. c. Suppose that P is a -system with P A, ad suppose that P is cotaied i ay other -system that cotais A. The P D T 2A P, with otatio as i (b). The miimal -system cotaiig A [which always exists] is also uique. Proof. (a) Suppose B; C 2 T 2A P. The B; C 2 P for every 2 A. Sice P is a -system, we have B\C 2 P for all 2 A. Cosequetly, B\C 2 T 2A P, i.e., T 2A P is a -system o. The aalogous statemet holds for rigs, -rigs, algebras ad -algebras. However, it fails for semirigs. A couterexample: let D f; 2; 3; 4g, A D f ; ; fg; f2; 3g; f4gg, ad A 2 D f ; ; fg; f2g; f3; 4gg. The A ad A 2 are semirigs b A \ A 2 D f ; ; fgg is ot. (b) Sice 2 2 fp W 2 Ag µ.a/, the family.a/ is oempty. It follows from (a) that T 2A P is a -system cotaiig A. Fially, if Q is a -system cotaiig A, the Q 2.A/, hece T 2A P Q. (c) Sice T 2A P is the -system geerated by A, we have T 2A P P ; sice P is cotaied i ay other -system that cotais A, we have P T 2A P...2 -System I Exercise.9 (..7). This exercise explores some equivalet defiitios of a -system. a. L is a -system iff L satisfies ( ), ( 0 2 ), ad ( 3). b. Every -system additioally satisfies ( 4 ), ( 5 ), ad ( 6 ). c. L is a -system iff L satisfies ( ), ( 0 2 ), ad ( 5). The coditios are: ( ) 2 L; ( 2 ) A 2 L H) A c 2 L; ( 0 ) A; B 2 L & A B H) B X A 2 L; 2 ( 3 ) For ay disjoit fa g L, S A 2 L; ( 4 ) A; B 2 L & A \ B D H) A [ B 2 L; ( 5 ) 8 fa g L, A "H) S A 2 L; ( 6 ) 8 fa g L, A #H) T A 2 L.

17 SECTION. -SYSTEMS, -SYSTEMS, AND SEMIRINGS 5 d. If a collectio L is oempty ad satisfies. 2 / ad. 3 /, the L is a -system. Proof. (a) Let L be a -system. The 2 L by ( ) ad ( 2 ). Suppose that A; B 2 L ad A B. The B c 2 L by (2) ad A \ B c D. By ( 3 ), B c [ A D B c [ A [ [ [ 2 L. By ( 2 ) agai, B X A D.B c [ A/ c 2 L. To show the iverse directio, we eed oly to show that ( ) ad ( 0 2 ) imply. 2 /: if A 2 L, the A c D X A 2 L. (b) Let L be a -system, so it satisfies ( ) ( 3 ) ad ( 0 2 ). To verify that ( 4) holds, first otice that D c 2 L. If A; B 2 L ad A \ B D, the A [ B D A [ B [ [ [ 2 L. To see that ( 5 ), let fa g L be icreasig. Let B D A ad B D A X A for > 2. The fb g L by ( 0 2 ) ad is disjoit. Hece, S A D F B 2 L. Fially, if fa g L is decreasig, the fa c g L is icreasig. Hece S A c 2 L by ( 5 ). The T A D. S A c /c 2 L. (c) If L is a -system, it follows from (a) ad (b) that ( 0 2 ) ad ( 5) hold. Now suppose that ( ), ( 0 2 ), ad ( 5) hold. It follows from the oly if part of (a) that ( ) ad ( 0 2 ) imply ( 2). To see ( 3 ) also hold, let fa g L be a disjoit sequece. We ca costruct a odecreasig sequece fb g by lettig B D S id A i. Notice that B 2 L for all. Hece, S B D S A, ad by ( 5 ), we have ( 3 ). (d) If L ad satisfies ( 2 ) ad ( 3 ), the there exists some A 2 L ad so D A [ A c 2 L by. 4 /. I Exercise.0 (..8). If L is a -system ad a -system, the S A 2 L wheever A 2 L for all 2 N. That is, L is closed uder coutable uios. Proof. This exercise proves that a -system which is S -stable is a -field (see Figure.). Let fa g L. Let B D A ad B D A c \ Ac 2 \ \ Ac \ A for all > 2. Sice L is a -system, fa c ; : : : ; Ac g L; sice L is a -system, k B 2 L. It follows from ( 3 ) that S A D S B 2 L. I Exercise. (..9). A -system is ot ecessarily a -system. Proof. For example, let D.0;. The followig collectio is a -system: L D f ; ;.0; =2 ;.=4; ;.=2; ;.0; =4 g : However, L is ot a -system because.0; =2 \.=4; D.=4; =2 L. I Exercise.2 (..0). Fid all -systems over D f! ;! 2 ;! 3 ;! 4 g with at least three elemets. Solio. ; ; f!i g; f! j ;! k ;!`g i j k ` ; ; f!i ;! j g; f! k ;!`g i j k `:

18 6 CHAPTER SET SYSTEMS I Exercise.3 (..). The collectio cosistig of ad the rsc itervals is ot a -system o R. Proof. This is ot a -system, b is a semirig. Cosider a otrival rsc iterval.a; b. Note that.a; b c D. ; a [.b; C/ is ot a rsc iterval, ad so is ot i this collectio. I Exercise.4 (..2). Suppose that for each i a oempty idex set A, L is a -system over. a. The collectio T 2A L is a -system o. b. Suppose that A 2 is such that A is cotaied i each L, ad suppose that fl W 2 Ag is the exhaustive list of all the -system that cotai A. The T 2A L is a -system that cotais A. If Q is a -system o that cotais A, the T 2A L Q. The miimal -system geerated by A always exists. c. Let L deote a -system over with L A ad where L is cotaied i ay other -system also cotaiig A. The L D T 2A L, with otatio as i (b). Therefore, the -system geerated by A always exists ad is uique. Proof. (a) It is clear that 2 T 2A L. Suppose A 2 T 2A L, the A 2 L for ay 2 A. Hece, A c 2 L for ay 2 A. So A c 2 T 2A L, i.e., T 2A L is closed uder complemetatio. To see T 2A L is closed uder disjoit uios, let fa g T 2A L be a disjoit sequece. The fa g L for ay implies S A 2 L for ay, which implies that S A 2 T 2A L. (b) From (a) we kow T 2A L is a -system, ad sice A L, 8 2 A, we kow that A T 2A L ; hece, T 2A L is a -system that cotais A. T 2A L Q because Q 2 fl W 2 Ag. (c) Sice L is cotaied i ay other -system cotaiig A, ad T 2A L is such a -system, so L T 2A L. Sice L 2 fl W 2 Ag, so T 2A L L...3 Semirig I Exercise.5 (..3). Is A D f g [ f.0; x W 0 < x 6 g a semirig over.0;? Solio. A is ot a semirig o.0;. Take.0; x ad.0; y with x < y. The.0; y X.0; x D.x; y A sice x > 0 by defiitio. I Exercise.6 (..4). This exercise explores some alterative defiitios of a semirig. a. Some defie A to be a semirig iff A is a oempty -system such that both E; F 2 A ad E F imply the existece of a fiite collectio C 0 ; C ; : : : ; C 2 A with E D C 0 C C F ad C i X C i 2 A for i D ; : : : ;. This defiitio of a semirig is equivalet to our defiitio of a semirig.

19 SECTION. -SYSTEMS, -SYSTEMS, AND SEMIRINGS 7 b. Some defie A to be a semirig by stipulatig (SR), (SR2), ad the followig property: A; B 2 A implies the existece of disjoit A-sets C 0 ; C ; : : : ; C with B X A D S id0 C i. Note that here B X A is ot ecessarily a proper differece. If A is a semirig by this defiitio, the A is a semirig by our defiitio, b the coverse is ot ecessarily true. Proof. (a) We first show that (SR), (SR2), ad (SR3) imply the above defiitio. (SR) ad (SR2) imply that A is a oempty -system (sice 2 A). Let E; F 2 A ad E F. By (SR3) there exists disjoit D ; : : : ; D 2 A such that F X E D S id D i. Let C 0 D E ad C i D E [ D [ [ D i for i D ; : : : ;. The E D C 0 C C D F, ad C i X C i D D i 2 A. Now suppose (a) holds. (SR): Sice A is oempty, there exists E 2 A; sice E E, there exists a fiite collectio E D C 0 C C E, which implies that C 0 D C D D C, ad so C i X C i D 2 A. (SR2) holds trivially. (SR3): Let A; B 2 A ad A B. The by the assumptio, there exists a fiite collectio C 0 ; C ; : : : ; C 2 A with A D C 0 C C B, ad B D C X C 2 A. The fb i g id A is disjoit, ad [ [ A A D A [ 4.C i X C i / 5 D A [.B X A/ D B: id B i id (b) Some ahors do apply this defiitio, for example, see Alipratis ad Border (2006); Dudley (2002). The proof is obvious. I Exercise.7 (..5). Let A cosist of as well as all rsc rectagles.a; b. The collectio of all fiite disjoit uios of A-sets is a semirig over R k. Proof. We prove a more geeral theorem. See Bogachev (2007, Lemma.2.4, p.8). For ay semirig, the collectio of all fiite uios of sets i forms a rig R. Proof. It is clear that the class R admits fiite uios. Suppose that A D S id A ad B D S k j D B k, where A i ; B i 2. The we have A \ B D S i6;j 6k A i \ B j 2 R. Note that A i \ B j 2 A, 8 i 2 f; : : : ; g ad j 2 f; : : : ; kg, sice a semirig is T -stable. Hece R admits fiite itersectios. I additio, 0 [ k[ A X B i X [ k\ A D A i X B j : id j D B j id j D Sice the set A i X B j D A i X A i \ B j is a fiite uio of sets i, oe has A i XB j 2 R. Furthermore, T k j D A T i X B j 2 because is -stable. Fially, the fiite list A i X B j is disjoit; hece, A X B is a fiite disjoit i2f;:::;g;j 2f;:::;kg uio of sets i.

20 8 CHAPTER SET SYSTEMS Now, sice A is a semirig [which is a well kow fact], we coclude that the collectio of all fiite disjoit uios of A-sets is a rig over R k [a rig is a semirig, see Exercise.22 (p.0)]. I Exercise.8 (..6). A arbitrary itersectio of semirigs o is ot ecessarily a semirig o. Solio. Ulike the other kids of classes of families of sets (e.g., Exercise.8 ad Exercise.2), the itersectio of a collectio of semirigs eed ot be a semirig. For example, let D f0; ; 2g, A D f ; ; f0g; fg; f2gg, ad A 2 D f ; ; f0g; f; 2gg. The A ad A 2 are semirigs (i fact, A 2 is a field), b their itersectio A D A \A 2 D f ; ; f0gg is ot a semirig as Xf0g D f; 2g is ot a disjoit uio of sets i A. Geerally, let A ad A 2 be two semirigs, ad 2 A ad 2 A 2. The 2 A \A 2, ad which meas that the complemet of every elemet i A \A 2 should be expressed as fiite uio of disjoit sets i A \A 2. As we have see i the example, this is a demadig requiremet. Of course, there is o pre-requiremet that should be i a semirig. See the ext Exercise.9. I Exercise.9 (..7). If A is a semirig over, must 2 A? Solio. Not ecessarily. I face, the simplest example of a semirig (a rig, a -rig) is just f g. I Exercise.20 (..8). Let A deote a semirig. Pick 2 N, ad let A; A ; : : : ; A 2 A. The there exists a fiite collectio fc ; : : : ; C m g of disjoit A-sets with A X S id A i D S m j D C j. Proof. Whe D, write AXA D AX.A \ A / ad ivoke (SR3). Now assume that the result is true for 2 N. Cosider C. 0 0 C [ [ m[ m[ A X A i X A X A C A X A C D C j X A C : id id A i Now for each j, there exists disjoit sets fd j ; : : : ; Dj k j g A such that j D C j j D C j X A C D The fdr j W j D ; : : : ; m; r D ; : : : ; m j g is a fiite pairwise disjoit subset of A, ad C m [ m[ [ j A X A i D Dr j : id k j [ rd D j r : j D rd

21 SECTION. -SYSTEMS, -SYSTEMS, AND SEMIRINGS 9 I Exercise.2 (..9). Other books deal with a system called a rig. We will ot deal with rigs of sets i this text, b sice the reader might refer to other books that deal with rigs, it is worthy to discuss the cocept. A collectio R of subsets of a oempty set is called a rig of subsets of iff (R) R, (R2) A; B 2 R implies A [ B 2 R, ad (R3) A; B 2 R implies A X B 2 R. That is, a rig is a oempty collectio of subsets closed uder uios ad differeces. a. is i every rig. b. R is a rig iff R satisfies (R), (R2), ad (R4) A; B 2 R with A B implies B X A 2 R. c. Every rig satisfies (R5) A; B 2 R implies A B 2 R. d. Every rig is a -system. e. Every rig is closed uder fiite uios ad fiite itersectios. f. R is a rig iff R a oempty -system that satisfies (R4) alog with (R6) A; B 2 R ad A \ B D imply A [ B 2 R. g. R is a rig iff R is a oempty -system that satisfies (R5). h. Suppose that fr W 2 Ag is the exhaustive list of all rigs that cotai A. The T 2A R is a rig that cotais A, ad T 2A R is cotaied i ay rig that cotais A. The miimal rig cotaiig A is always exists ad is uique. i. The collectio of fiite uios of rsc itervals is a rig o R. j. Let be ucoutable. The collectio of all amc subsets of is a rig o. Proof. (a) By (R), there exists some set A 2 R, it follows from (R3) that D A X A 2 R. (b) We eed oly to prove that (R3) () (R4) uder (R) ad (R2). (R3) H) (R4) is obvious. (R4) H) (R3): Let A; B 2 R, ad ote that B X A D.B [ A/ X A 2 R sice A A [ B, ad B [ A 2 R by (R2).

22 0 CHAPTER SET SYSTEMS (c) Let A; B 2 R. By (R3), AXB 2 R, ad B XA 2 R; by (R2),.A X B/[.B X A/ 2 R. Observe that A B D.A X B/ [.B X A/, ad we complete the proof. (d) Let A; B 2 R. It is clear that A \ B D.A [ B/ X.A B/. Note that A [ B 2 R [by (R2)], A B 2 R [by (c)], ad.a [ B/ X.A B/ 2 R [by (R3)]. Therefore, A \ B 2 R ad R is a -system. (e) Just follows (R2) ad (d). (f) To see the oly if part, suppose R is a rig. The (d) meas that R is a oempty -system, (R3) H) (R4) [by part (b)], ad (R2) H) (R6) [by defiitio]. Now we prove the if part. Note that (R) (R2) By assumptio; Let A; B 2 R. We ca write A [ B as A [ B D.A X B/ [.B X A/ [.A \ B/ D A X.A \ B/ [ B X.A \ B/ [.A \ B/ : Now (R4) implies that A X.A \ B/ 2 R, ad B X.A \ B/ 2 R; (R6) implies that A [ B 2 R. 2 (R3) Let A; B 2 R. The A X B D [.A X B/ D.A \ A c / [.A \ B c / D A \.A c [ B c / D A \.A \ B/ c D A X.A \ B/. Clearly, A \ B A, so (R4) implies that A X B 2 R. (g) To see the oly if part, suppose that R is a rig. The (R) ad (d) implies R is a oempty -system, ad we have (R5) by (c). For the iverse directio, suppose that R satisfies the give assumptios. (R) R by assumptio; (R2) Let A; B 2 R. TheA [ B D.A B/ [.A \ B/ D.A B/.A \ B/. Sice R is a -system, A \ B 2 R. Thus, (R5) implies (R2). (R3) Let A; B 2 R. Note that A X B D.A B/ \ A. The (R5) implies that A B 2 R, ad.a B/ \ A 2 R sice R is a -system. 3 (h) Similar to Exercise.8 ad Exercise.2. (i) See Exercise.5 (p.47). (j) (R) is trivial. (R2) holds because every fiite (i fact, coutable) uio of amc sets is amc (see, e.g., Rudi 976). To see (R3), let A; B be amc. Sice A X B D A X.A \ B/ A, ad A \ B A, we kow that A X B is amc. I Exercise.22 (..20). This problem explores the relatioship betwee semirigs ad rigs. 2 For A X B D A X.A \ B/, see part (g) of this exercise 3 Vestrup (2003, p.6) hits that A X B D A.A \ B/.

23 SECTION. -SYSTEMS, -SYSTEMS, AND SEMIRINGS a. Every rig is a semirig. However, ot every semirig is a rig. b. Let A deote a semirig o, ad let R cosist of the fiite disjoit uios of A-sets. The R is closed uder fiite itersectios ad disjoit uios. c. If A; B 2 A ad A B, the B A 2 R. d. A 2 A, B 2 R, ad A B imply B A 2 R. e. A; B 2 R ad A B imply B A 2 R. f. R is the miimal rig geerated by A. g. A semirig that satisfies (R2) is a rig. Proof. (a) Let R be a rig. The (R) [R ] ad (R3) [R is closed uder differeces] imply that there exists A 2 R such that D AXA 2 R. Thus, (SR) is satisfied. To see that R satisfies (SR2) [R is a -system], refer Exercise.2 (d). Fially, (R4) [Exercise.2 (b)] implies (SR3). To see a semirig is ot ecessary a rig, ote that the collectio f ;.a; b j a; b 2 Rg is a semirig, b is ot a rig: let < a < b < c < d < C, the.a; b [.c; d. Note that a semirig is a rig if for ay A; B 2 we have A [ B 2 [Figure. (p.), ad part (g) of this exercise]. Ay semirig geerates a rig as i the Claim i Exercise.5 (p.47). (b) Let A be a semirig o, ad let 8 9 < [ = R A : i W A i 2 A ad 2 N ; : id To prove R is closed uder fiite itersectios, let A D S m j D A j, ad B D S kd B k, where the A j s are disjoit ad i A, as are the B k s. The 0 m[ A \ B A j 0 [ A B k A D m[ [ j D kd j D kd A j \ B k D [ 6j 6m 6k6 A j \ B k hi 2 R; where hi holds because the A j \ B k s are disjoit ad i A [by (SR2)]. Sice the itersectio of ay two sets i R is i R, it follows by iductio that so is the itersectio of fiitely may sets i R. A disjoit uio of fiitely may sets i R is clearly i R. (c) Let A; B 2 A ad A B. The by (SR3), there exists disjoit C ; : : : ; C k 2 A with B X A D S k id C i. Thus, B X A 2 R by defiitio.

24 2 CHAPTER SET SYSTEMS (d) Let A 2 A, B 2 R, ad A B. The, B 0 [ A h2i A [ A D.A i X A/ D id A i id [ id Ai.A i \ A/ h3i 2 R; where h2i follows the fact that B 2 R [the A i s are i A ad disjoit], ad h3i follows part (c) i this problem [ote that A i 2 A; A 2 A, ad by (SR2), A i \ A 2 A]. (e) Let A; B 2 R ad A B. The [ m[ [ B X A A A j A 6 m[ D 4B k B k kd j D kd j D kd j D A j 3 A7 5 D 2 [ m\ 4 3 B k X A j 5 : Note that B k ; A j 2 A, the B k \ A j 2 A [(SR2)], ad by part (c), B k X A j D B k B k \ A j 2 R: Furthermore, by part (b), T m j D B k X A j 2 R, ad so B A 2 R. (f) Let <.A/ be the class of rigs cotaiig A, ad let C 2 <.A/. By defiitio, if A 2 R, the A D S id A i, where fa i g id A are disjoit. The A 2 C sice C is a rig cotaiig A. Hece, R is the miimal rig cotaiig A. (g) Let A be a semirig satisfyig (R2) [A; B 2 A H) A [ B 2 A]. The A is oempty sice 2 A by defiitio of a semirig. By (R2), A is S -stable; hece, to prove A is a rig, we eed oly to prove that A is closed uder differece. Let A; B 2 A. The A X B D A k[.a \ B/ D C i 2 A; id where fc i g k id A are disjoit, ad equality () follows (SR3). I Exercise.23 (..2). Let be ifiite, ad let A 2 have 0. We will show that the rig geerated by A has 0. a. Give C 2, let C deote the collectio of all fiite uios of differeces of C-sets. If card.c/ 0, the card.c / 0. Also, 2 C implies C C. b. Let A 0 D A, ad defie A D A for >. The A S D0 A <.A/, where <.A/ is the miimal rig geerated by A ad where [witho loss of geerality] 2 A. Also, card. S D0 A / 0. c. S D0 A is a rig o, ad from the fact that <.A/ is the miimal rig cotaiig A, we have S D0 A D <.A/, ad thus card.<.a// 0. d. We may geeralize: if A is ifiite, the card.a/ D card.<.a//.

25 SECTION. -SYSTEMS, -SYSTEMS, AND SEMIRINGS 3 Proof. (a) Let C 0 fc i X C j W C i ; C j 2 Cg. Sice card.c/ 0 [C is coutable], we ca write C as C D fc g : We ow show that card.c 0 / 0. Notice that for ay C 2 C, we ca costruct a bijectio o N oto C X C fc X C i W C i 2 Cg as follows f C.i/ D C X C i ; b which meas that C X C is coutable. The, C 0 D [ ŒC X C C 2C is a coutable uio of coutable sets, so it is coutable [uder the Axiom of Choice, see (Hrbacek ad Jech, 999, Corollary 3.6, p. 75)]. Now we show that for ay 2 N, the set C defied by 9 8 < C D : [ = Ci 0 W C i 0 2 C 0 ad Ci 0 C j 0 wheever i j ; id is coutable. We prove this claim with the Iductio Priciple o 2 N. Clearly, this claim holds with D sice i this case, C D C 0. Assume that it is true for some 2 N. We eed to prove the case of C. However, where C C D C [ xc 0 ; xc 0 C 0 2 C 0 W C 0 C 0 i 8 i 6 : Because C 0 is coutable, we coclude that xc 0 C 0 is amc. Therefore, C C is coutable. Hece, by the Iductio Priciple, card.c / 0 for ay 2 N, ad C D [ 2N C (.3) is coutable. We ow show that if 2 C, the C C. Let C 2 C, the C 2 C 0 because C D C X ; therefore, C C 0 C : [Remember that C 0 D C ad (.).] (b) By the defiitio of A, we kow A D A, the collectio of all fiite uios of differeces of A sets. Sice 2 A, we kow from part (a) that A A D A ; therefore, [ A A : (.4) D0

26 4 CHAPTER SET SYSTEMS We are ow ready to prove that S D0 <.A/. We use the Iductio Priciple to prove that A i <.A/; 8 i 2 N: (Pi) Clearly, P0 holds as A 0 D A <.A/. Now assume P holds. We eed to prove P C. Notice that A C D A, the collectio of all fiite uios of differeces of A -sets, we ca write a geeric elemet of A C as A C D m[ Aj 0 ; where Aj 0 D A0 X A00, ad A0 ; A00 2 A. Sice A <.A/ by P, we kow that A j D A 0 X A00 2 <.A/ by (R3); therefore, A C D S m j D A0 j 2 <.A/ by (R2). This proves P C. The, by the Iductio Priciple, we kow that A <.A/, 8 2 N; therefore, [ A <.A/: (.5) Combie (.2) ad (.5) we have A D0 j D [ A <.A/: (.6) D0 To prove card. S D0 A / 0, we first use the Iductio Priciple agai to prove that A is coutable, 8 2 N. Clearly, A D A is coutable by part (a). Assume A is coutable, the A C D A is coutable by part (a) oce agai. Therefore, S D0 A is coutable [uder the Axiom of Choice]. (c) Clearly, S D0 A za, so (R) is satisfied. To see (R2) ad (R3), let A; B 2 za. The there exist m; 2 N such that A 2 A m ad B 2 A. We have show i part (a) that A C D A A [alog with the Iductio Priciple]. Therefore, either A m A [if m 6 ] or A A m [if 6 m]. Witho loss of geerality, we assume that m 6, i.e., A m A ; therefore, A 2 A m H) A 2 A. Therefore, A; B 2 A implies that A [ B D.A X / [.B X / 2 A D A C za; [this proves (R2)], ad 0 A X B [ A A i id 0 A [ B B j j D A D A [ C B id [ A i X B j 2 A A ; [this proves (R3)]. Hece, za is a rig, ad za D <.A/; furthermore, we have card.<.a// 0. (d) Straightforward.

27 SECTION.2 FIELDS 5.2 Fields I Exercise.24 (.2.). The collectio F D fa W A is fiite or A c is fiiteg is a field o. Proof. 2 F because c D is fiite; let A 2 F. If A is fiite, A c 2 F as.a c / c D A is fiite; if A c is fiite A c 2 F. Thus, F is closed uder complemets. Fially, let A; B 2 F. There are two cases: (i) both A ad B are fiite, the A[B is fiite, whece A [ B 2 F ; (ii) at least oe of A c or B c is fiite. Assume that B c is. We have.a [ B/ c D A c \ B c B c, ad thus.a [ B/ c is fiite, so that gai A [ B 2 F. I Exercise.25 (.2.2). Let F 2 be such that 2 F ad A X B 2 F wheever A; B 2 F. The F is a field o. Proof. We eed to check F satisfies (F) (F3). 2 F by assumptio. Let A D ad B 2 F. The B c D X B 2 F. Let A; B 2 F. The A c ; B c 2 F. Sice.A [ B/ c D A c \ B c D A c X B 2 F, we must have A [ B D.A [ B/ c c 2 F. I Exercise.26 (.2.3). Every -system that is closed uder arbitrary differeces is a field. Proof. We oly eed to show that it is closed uder fiite uios, ad it comes from the previous exercise. I Exercise.27 (.2.4). Let F 2 satisfy (F) ad (F2), ad suppose that F is closed uder fiite disjoit uios. The F is ot ecessarily a field. Solio. For example, let D f; 2; 3; 4g, ad F D ; ; f; 2g ; f3; 4g ; f2; 3g ; f; 4g : F satisfies all the requiremets, b which is ot a field sice, for example, f; 2g [ f2; 3g D f; 2; 3g F : I Exercise.28 (.2.5). Suppose that F F 2 F 3, where F is a field o for each 2 N. The S F is a field o. Proof. (F) 2 F, for each 2 N, so 2 S F ; [Of course, it is eough to check that 2 F for some F.] (F2) Suppose A 2 S F. The there exist 2 N such that A 2 F. So A c 2 F H) A c 2 S F ; (F3) Let A; B 2 S F. The 9 m 2 N such that A 2 F m, ad 9 2 N such that B 2 F. Hece, A [ B 2 F m [ F S F. I Exercise.29 (.2.6). The collectio cosistig of R k,, ad all k-dimesioal rectagles of all forms fails to be a field o R k.

28 6 CHAPTER SET SYSTEMS Solio. Cosider k D ad Œa; b, where a; b 2 R. The Œa; b c D. ; a/ [.b; C/ is ot a iterval. The k > 2 case ca be geeralized easily. For example, let A D k id Œa i ; b i : The A c is ot a rectagle. I Exercise.30 (.2.7). The collectio cosistig of ad the fiite disjoit uios of k-dimesioal rsc subrectagles of the give k-dimesioal rsc rectagle.a; b is a field o. Proof. A more geeral propositio ca be foud i Follad (999, Propositio.7). Deote the set system give i the problem as, a semirig, ad the collectio of ad the fiite disjoit uios of k-dimesioal rsc subrectagles as A First D S i2 I i by defiitio, where I i 2. If A; B 2 ad B c D S id C i, where C i 2. The A X B D S id.a \ C i/ ad A [ B D.A X B/ [ B. Hece A X B 2 A ad A [ B 2 A. It ow follows by iductio that if A ; : : : ; A 2, the S id A i complemets. 2 A. It is easy to see that A is closed uder I Exercise.3 (.2.8). A arbitrary itersectio of fields o is a field o. Proof. Let ff W 2 Ag be a set of fields o, where A is some arbitrary set of idexes. The (F) 2 T 2A F sice 2 F for ay 2 A. (F2) Let B 2 T 2A F, the A c 2 F, for ay 2 A; hece A c 2 T 2A F. (F3) Let B; C 2 T 2A F. The B; C 2 F, 8 2 A. Hece, B[C 2 F, 8 2 A, ad B \ C 2 T 2A F. I Exercise.32 (.2.9). Let be arbitrary, ad let A 2. There exists a uique field F o with the properties that (i) A F, ad (ii) if G is a field with A G, the F G. This field F is called the [miimal] field [o ] geerated by A. Proof. Let ff W 2 Ag be the exhaustive set of fields o cotaiig A. The T 2A F is the desired field. I Exercise.33 (.2.0). Let A ; : : : ; A be disjoit. What does a typical elemet i the miimal field geerated by fa ; : : : ; A g look like? Solio. Refer to Ash ad Doléas-Dade (2000, Exercise.2.8). To save otatio, let F deote the miimal field geerated by A fa ; : : : ; A g. We cosider a elemet of F X f ; g. We ca write a typical elemet B 2 F as follows,

29 SECTION.2 FIELDS 7 B D B B 2 B m ; where is a set operatio either [ or \, ad B i 2 A ; : : : ; A ; A c ; : : : ; Ac for each i 2 f; : : : ; mg. I Exercise.34 (.2.). Let S be fiite, ad deote the set of sequeces of elemets of S. For each! 2, write! D z.!/ ; z 2.!/ ; : : : ; so that z k.!/ deotes the k-th term of! for all k 2 N. For 2 N ad H S, let C.H / f! 2 j z.!/ ; : : : ; z.!/ 2 H g : Let F C.H / ˇˇ 2 N; H S : The F is a field of subsets of S. [The sets C.H / are called cyliders of rak, ad F is collectio of all cyliders of all raks.] Proof. See Remark.2 (p.2) for more details abo Cyliders. To prove F is a field, ote that (F) 2 F. Cosider C.S /; the! 2 C.S /, 8! 2, which meas C.S /. Hece, D C S 2 F : (F2) To prove that F is closed uder complemets, cosider ay C.H / 2 F. By defiitio, C.H / f! 2 j z.!/ ; : : : ; z.!/ 2 H g : The, C.H / c D! 2 W Œz.!/; : : : ; z.!/ H D! 2 W Œz.!/; : : : ; z.!/ 2 H c D C H c 2 F : (-system) Fially, we eed to prove F is closed uder fiite itersectios. 4 4 It is hard to prove that F is closed uder fiite uios. See below for my first b failed try. (Wrog!) Let C m.g/ ; C.H / 2 F, where m; 2 N ad G S m ; H S. By defiitio, C m.g/ [ C.H / D! 2 ˇ z.!/; : : : ; z m.!/ o [ 2 G! 2 ˇˇ Œz.!/; : : : ; z.!/ 2 H D! 2 ˇ z.!/; : : : ; z m^.!/ o 2.H [ G/ 2 D C m^.g m^ [ H m^ / 2 F ;

30 8 CHAPTER SET SYSTEMS Cosider two cyliders, C m.g/ ad C.H /, where m; 2 N, G S m, ad H S. We eed to prove that C m.g/ \ C.H / 2 F. I fact, C m.g/ \ C.H / D C m_.g m^ \ H m^ / G m.m^/ [ H.m^/ 2 F ; where, for example, G m^ i equality (2), G m^ S m^, G m.m^/ S m.m^/, ad G m^ G m.m^/ D G. To see why equality () holds, we eed the followig facts: Claim. Suppose that m 6, H D G H m, ad G S m. The C m.g/ C.H /. Proof. Pick ay! 0 2 C.H /. By defiitio, h z! 0 ; : : : ; z! 0i 2 H D G H m ; which meas that h z! 0 ; : : : ; z m! 0i 2 G H)! 0 2 C m.g/: Claim 2. If G H S, the C.G/ C.H /. Proof. Straightforward. Claim 3. For ay m; 2 N, ad G S m ; H S, we have C m.g/ [ C.H / C m^.g m^ [ H m^ /: Proof. Witho loss of ay geerality, we assume that m ^ D m. Pick ay! 0 2 C m.g/ [ C.H /. The, h z.! 0 /; : : : ; z m! 0i h 2 G; or z.! 0 /; : : : ; z! 0i 2 H: From Claim 2, we have h z.! 0 /; : : : ; z m! 0i 2 G [ H m ; or h z.! 0 /; : : : ; z! 0i 2.G [ H m / H m ; where H m S m, ad H D H m H m. The, by Claim, if m ^ D m, we have! 0 2 C m.g [ H m / : Claim 4. For ay m; 2 N, ad G S m ; H S, we have C m.g/ [ C.H / C m^.g m^ [ H m^ /: Proof. We still assume that m ^ D m. Pick ay! 0 2 C m^.g m^ [ H m^ / D C m.g [ H m /. By defiitio, h z! 0 ; : : : ; z m! 0i 2 G [ H m I that is, h z! 0 ; : : : ; z m! 0i 2 G or h z! 0 ; : : : ; z m! 0i 2 H m :

31 SECTION.2 FIELDS 9 where G m^ ; H m^ S m^, G m.m^/ S m.m^/, H.m^/ S.m^/, G D G m^ G m.m^/, H D H m^ H.m^m/, ad we defie G 0 D H 0 D. I Exercise.35 (.2.2). Suppose that A is a semirig o with 2 A. The collectio of fiite disjoit uios of A-sets is a field o. [Compare with Example 3 ad Exercise.30.] Proof. Let A be a semirig, ad 2 A. Let F be the collectio of fiite disjoit uios of A-sets, tha is, A 2 F iff for some 2 N we have A D S id A i, where A i s are disjoit A-sets. F is a field: (i) 2 F sice D [ 2 F. (ii) Let A 2 F. The A D S id A i, where 2 N ad fa i g id A. To prove A c 2 F, we eed oly to prove A c i 2 F sice A c D T id Ac i, ad A is a semirig [ T -stable]. B A c i 2 F is directly from (SR3) ad the fact that 2 A sice A c i D X A i D S o i i j D C j i, where Cj i A is disjoit, ad j D i 2 N, 8 i 2 f; : : : ; g, that is, each A c i is a fiite disjoit uio of A-sets. Thus, F is closed uder complemets. Istead of provig that F satisfies (F3) directly, we prove that F is a - system. Let B ; B 2 2 F. The [ k[ [ k[ B \ B 2 A i A A j A D 4 A i \ A j 5 D [ A i \ A j : id j D id j D i;j Note that A i \ A j 2 A by (SR2). Hece B \ B 2 2 F. I Exercise.36 (.2.3). Let f W! 0. Give A 0 2 0, let f.a 0 / D ff.a 0 / W A 0 2 A 0 g, where f.a 0 / is the usual iverse image of A 0 uder f. a. If A 0 is a field o 0, the f.a 0 / is a field o. b. f.a/ may ot be a field over 0 eve if A is a field o. Proof. (a) Let A 0 be a field o 0. (i) Sice D f. 0/ ad 0 2 A 0, we have that 2 f.a 0 /. (ii) If A 2 f.a 0 /, the A D f.a 0 / for some A 0 2 A 0. Therefore, A c D Œf.A 0 / c D f..a 0 / c /, ad.a 0 / c 2 A 0 sice A 0 is a field. It follows that A c 2 f.a 0 /, so that f.a 0 / is closed uder complemets. (iii) To see that f.a 0 / is closed uder fiite uios, let fa i g id A, where 2 N. Therefore, for each i 2 f; : : : ; g, there is A 0 i 2 A 0 with A i D f.a i /. Therefore, 0 [ [ A i D f A 0 [ i D A 0 A i 2 f.a 0 /; id sice S id A0 i 2 A0. id id

32 20 CHAPTER SET SYSTEMS (b) The simplest case is that f is ot oto [surjective]. I this case, f. / 0; that is, 0 A 0, ad so A 0 is ot a field o 0. I Exercise.37 (.2.4). Let be ifiite, ad let A 2 have 0. Let f.a/ deote the miimal field geerated by A [Exercise.32]. We will show that card.f.a// 0. a. Give a collectio C, let C deote the collectio of i. fiite uios of C-sets, ii. fiite uios of differeces of C-sets, ad iii. fiite uios of complemets of C-sets. If 2 C, the C C. If card.c/ 0, the card.c / 0. Proof. I Exercise.38 (.2.5). Some books work with a system of sets called a algebra. A algebra o is a oempty collectio of subsets of that satisfies (F2) ad (F3). a. F is a algebra o iff F is a rig o with 2 F. b. F is a algebra iff F is a field. Thus algebra ad field are syoymous. Proof. (a: H)) Suppose F is a algebra. The, (R) (R2) F by assumptio. F is S -stable follows (F3). (R3) The assumptio of 2 F ad (F2) imply that if A; B 2 F, the A c D A 2 F ad B c D B 2 F. The A c [ B (F3) (F2) h 2 F H) A c [ B i c 2 F H) ŒA X B 2 F : This proves that F is closed uder differece. (a: (H) Suppose F is a rig ad 2 F. To prove F is a algebra o, ote that (A) F sice F is a rig. (F2) Let A 2 F. Because 2 F ad (R3), we have A c D A 2 F. This proves that F is closed uder differece. (F3) S -stability follows (R2).

33 SECTION.3 -FIELDS 2 (b) We eed oly to prove that F is a field if F is a algebra sice the reverse directio is trivial. Suppose F is a algebra. We wat to show 2 F. Sice F by defiitio of a algebra, there must exist A 2 F. The A c 2 F by (F2), ad so D A[A c 2 F by (F3)..3 -Fields I Exercise.39 (.3.). A collectio F of sets is called a mootoe class iff (MC) for every odecreasig sequece fa g of F -sets we have S A 2 F, ad (MC2) for every oicreasig sequece fa g of F -sets we have T A 2 F. a. If F is both a field ad a mootoe class, the F is a -field. b. A field is a mootoe class if ad oly if it is a -field. Proof. See Chug (200, Theorem 2..). a. Let F is both a field ad a MC. Let fa g F, the B D S id A 2 F sice F is a field, B B C, ad S A D S B 2 F. b. We oly eed to show the IF part. B it is trivial: A -filed is a field ad a MC. I Exercise.40 (.3.2). This problem discusses some equivalet formulatios of a -field. a. F satisfies (S), (S2), ad closure uder amc itersectios iff F is a -field. b. Every field that is closed uder coutable disjoit uios is a -field. c. If F satisfies (S), closure uder differeces, ad closure uder coutable uios or closure uder coutable itersectios, the F is a -field. Proof. (a) For the ONLY IF part, let fa g F ad F satisfy (S) ad (S2). The A c 2 F for ay 2 N; hece, S A D T c Ac 2 F. The IF part is proved by the same logic. (b) We eed oly to prove F is closed uder coutable uitios. Let F be a field, ad fa g F. Let 0 [ B k D A k It is clear that fb g F is disjoit, ad S B D S A. This completes the proof. id A i A c :

34 22 CHAPTER SET SYSTEMS (c) We oly eed to prove (S2), that is, F is closed uder complemetatio. Let A 2 F. By (S), 2 F, the A c D X A 2 F sice by assumptio, F is closed uder differece. I Exercise.4 (.3.3). Prove the followig claims. a. A fiite uio of -fields o is ot ecessarily a field o. b. If a fiite uio of -fields o is a field, the it is a -field as well. c. Give -fields F F 2 o, it is ot ecessarily the case that S F is a -field. Proof. (a) Let ff i g id be a class of -fields, ad cosider S id A i, where A i 2 F i. Note that it is possible that S id A i F j for ay j, so S id A i S id F i. For example (Athreya ad Lahiri, 2006, Exercise.5, p.32), let D f; 2; 3g ; F D fg ; f2; 3g ; ; ; F 2 f; 2g ; f3g ; ; : It is easy to verify that F ad F 2 are both -fields, b F [ F 2 is ot a field sice fg [ f3g D f; 3g F [ F 2. (b) Witho loss of ay geerality, we here just cosider two -fields, F ad F 2, o. Cosider a sequece fa g F [ F 2. The we ca costruct two sequeces, oe i F ad oe i F 2. Particularly, the sequece of sets A F is costructed as follows: 8 < A D A ; if A 2 F : ; otherwise. The sequece of sets A 2 F2 is costructed similarly. The S kd A k 2 F ad S md A2 m 2 F 2 sice both F ad F 2 are -fields, ad 0 0 [ [ [ A A A : If F [ F 2 is a field, we have 0 [ [ A A A 0 [ A A 2 A 2 A 2 F [ F 2 : (c) See Broughto ad Huff (977) for a more geeral result. Let D N ad for all 2 N, let F D fg ; : : : ; fg : Sice fg ; : : : ; fmg fg ; : : : ; fg whe m <, we have F F 2. It it clear that fg ; f2g ; : : : 2 S F, b

35 SECTION.3 -FIELDS 23 [ [ fg D f; 2; : : :g F ; sice there does ot exist a F such that f; 2; : : :g 2 F, for ay 2 N. I Exercise.42 (.3.5). A subset A R is called owhere dese iff every ope iterval I cotais a ope iterval J such that J \ A D. Clearly ad all subsets of a owhere dese set are owhere dese. A subset A R is called a set of the first category iff A is a coutable uio of owhere dese sets. a. A amc uio of sets of the first category is of the first category. b. Let F D A R W A or A c is a set of the first category. The F is a -field of subsets of R. Proof. Refer Gameli ad Greee (999, Sectio.2) for the more detailed defiitios ad discussio of owhere dese ad the first category set. (a) Cosider a coutable sequece of sets of the first category, fa g. The A D S id A i for ay 2 N, where A i are owhere dese. Clearly, the id amc uios of amc uios is still amc, which proves the claim. (b) Let F D A R W A or A c is a set of the first category. The 2 F sice is of the first category ad D c. To see F is closed uder complemetatio, let A 2 F. (i) If A is of the first category, the A c 2 F sice.a c / c D A is of the first category; (ii) If A c is of the first category, the A c 2 F by the defiitio of F. I ay case, A 2 F implies that A c 2 F. Fially, to see F is - S -stable, let fa g be a sequece of F -sets. There are two cases: (i) Each A is of the first category. The part (a) of this exercise implies that S A 2 F. (ii) Some A c is of the first category. I this case, we assume witho loss of geerality that A c is of the first category, ad we S have that A c T D Ac Ac. It is trivial that S A c is of the first category sice A c is, ad every subset of the first category is of the first category. Particularly, let A c D S B, where the B s are owhere dese sets. Sice S A c A c, we must ca rewrite S A c as 0 c [ A D C ; A where every C is a subset of B ad some C s maybe be empty. Note that the S every C is owhere dese o matter C D or ot. Cosequetly, A c is of the first category by defiitio. I Exercise.43 (.3.6). A -rig of subsets of is a oempty collectio of subsets of that is closed uder differeces as well as coutable uios. a. Every -rig is closed uder fiite uios ad amc itersectios.

36 24 CHAPTER SET SYSTEMS b. F is a -field iff F is a -rig with 2 F. c. State ad prove a existece ad uiqueess result regardig the [miimal] -rig geerated by a collectio A of subsets of. Proof. (a) Let R be a -rig. We first prove that 2 R. Sice R, there exists A 2 R; moreover, sice R is closed uder differece, we have D A X A 2 R. Now cosider a arbitrary sequece of R-sets A ; : : : ; A ; ; ; : : :. Because R is - S -stable, we kow that [ A i D.A [ A 2 [ [ A / [. [ [ / 2 R; id which proves that R is S -stable. To see R is closed uder amc itersectios, let fa g S A 2 R. Let A 0 D A X A ; 8 2 N: The A 0 R, S A0 2 R, ad 0 [ A A D A 0 \ A 2 R R. The A D sice A X S A0 2 R. [Basically, I let A be the uiversal space, ad A 0 be the complemets of A i A.] (b) Suppose that F is a -field. The 2 F be (S). To see F is closed uder differece, let A; B 2 F. The (S2) implies that B c 2 F. Sice F is T -stable, we have A X B D A \ B c 2 F. The fact that F is - S -stable follows (S3). Now suppose that F is a -rig with 2 F. We eed oly to prove that F satisfies (S2). Let A 2 F. Sice 2 F ad F is closed uder differece, we have A c D X A 2 F : (c) Stadard. Omitted. I Exercise.44 (.3.9). a. If A A 0.A/, the.a 0 / D.A/. b. For ay collectio A 2,.A/.A/.A/. c. If the oempty collectio A is fiite, the.a/ D f.a/. d. For arbitrary collectio A, we have.a/ D f.a/. e. For arbitrary collectio A, we have f.a/ D f.a/. Proof. (a) O the first had, A A 0 implies that.a/.a 0 /; o the secod had, A 0.A/ implies that.a 0 /.A/ D.A/. We thus get the equality.

37 SECTION.3 -FIELDS 25 (b) Let.A/,.A/, ad.a/ deote the collectio of all -systems, - systems, ad -fields of subsets of that cotai A, respectively. With this, we may defie.a/ D \ P ;.A/ D \ L; ad.a/ D \ F : P 2.A/ L2.A/ F 2.A/ It is easy to see that ay -field cotaiig A is a -system cotaiig A; hece.a/.a/, ad so.a/.a/. (c) It is clear that f.a/.a/; sice 0 < jaj <, the field f.a/ is fiite ad so it is a -field. The.A/ f.a/. (d) O the first had, f.a/.a/ implies that f.a/.a/ D.A/. O the secod had, A f.a/ ad so.a/ f.a/. (e) It follows from (d) that f.a/ D.A/. By defiitio, f.a/ is the miimal field cotaiig.a/. B.A/ itself is a field; hece f.a/ D.A/. I Exercise.45 (.3.6). Let F D.A/, where A 2. For each B 2 F there exists a coutable subcollectio A B A with B 2.A B /. Proof. Let B D B 2 F W 9A B A such that A B is coutable ad B 2.A B / : (.7) It is clear that B F. For ay B 2 A, take A B D fbg; the A B D fbg is coutable ad B 2 fbg D f ; ; B; B c g; hece A B. We ow show that B is a -field. Obviously, 2 B sice 2 F ad 2 f g D f ; g. If B 2 B, the B 2 F ad there exists a coutable A B A such that B 2.A B /; b which mea that B c 2 F ad B c 2.A B /, i.e., B c 2 B. Similarly, it is easy to see that B is closed uder coutable uios. Thus, B is a -field cotaiig A, ad so F B. We thus proved that B D F ad the get the result. I Exercise.46 (.3.8). Give A 2 ad B, let A \ B D fa \ B W A 2 Ag ad let.a/ \ B D fa \ B W A 2.A/g. a..a/ \ B is a -field o B. b. Next, defie B.A \ B/ to be the miimal -field over B geerated by the class A \ B. The B.A \ B/ D.A/ \ B. Proof. This claim ca be foud i Ash ad Doléas-Dade (2000, p. 5). (a) B 2.A/ \ B as 2.A/. If C 2.A/ \ B, the C D A \ B with A 2.A/; hece B XC D A c \B 2.A/\B. To see that.a/\b is closed uder coutable uios, let fc g.a/ \ B. The each C D A \ B with A 2.A/. Hece,

38 26 CHAPTER SET SYSTEMS 0 [ [ [ C D.A \ B/ A A \ B 2.A/ \ B: (b) First, A.A/, hece A\B.A/\B. Sice.A/\B is a -field o B by (a), we have B.A \ B/.A/ \ B. To establish the reverse iclusio we must show that A \ B 2 B.A \ B/ for all A 2.A/. We use the good sets priciple. Let G D fa 2.A/ W A \ B 2 B.A \ B/g : We ow show that G is a -field cotaiig A. It is evidet that 2 G. If A 2 G, the A\B 2 B.A \ B/ ad A 2.A/; hece, A c \B D B X.A \ B/ 2 B.A \ B/ implies that A c 2 G. To see G is closed uder coutable uios, let fa g G with A \ B 2 B.A \ B/ for all 2 N. The 0 [ A \ B D.A \ B/ 2 B.A \ B/ : A Sice A G, we have.a/ G ; hece.a/ D G : every set i.a/ is good. I Exercise.47 (.3.9). Suppose that A D fa ; A 2 ; : : :g is a disjoit sequece of subsets of with S A D. The each.a/-set is the uio of a at most coutable subcollectio of A ; A 2 ; : : :. Proof. Let C D A 2.A/ W A is a at most coutable uio of A-sets : It is easy to see that 2 C sice D S A. If A 2 C, the A D S i2j A i, where J is at most coutable. Hece A c D S A S X i2j A i is a at most coutable uio of A-sets, that is, C is closed uder complemets. It is also easy to see that C is closed uder coutable uios ad A C. Hece, C is a -field ad.a/ D C. I Exercise.48 (.3.20). Let P deote a -system o, ad let L deote a -system o with P L. We will show that.p / L. Let.P / deote the -system geerated by P, ad for each subset A we defie G A D fc W A \ C 2.P /g. Proof. See Vestrup (2003, Claim, p. 82). I Exercise.49 (.3.2). Let F deote a field o, ad let M deote a mootoe class o [See Exercise.39]. We will show that F M implies that.f / M. Let m.f / deote the miimal mootoe class o geerated by F. That is, m.f / is the itersectio of all mootoe classes o cotaiig the collectio F. a. To prove the claim, it is sufficiet to show that.f / m.f /.

39 SECTION.3 -FIELDS 27 b. If m.f / is a field, the.f / m.f /. c. 2 m.f /. d. Let G D fa W A c 2 m.f /g. G is a mootoe class o ad m.f / G. e. m.f / is ideed closed uder complemets. f. Let G D A W A [ B 2 m.f / for all B 2 F. The G is a mootoe class such that F G ad m.f / G. g. Let G 2 D B W A [ B 2 m.f / for all A 2 m.f /. The G 2 is a mootoe class such that F G 2, ad m.f / G 2. h. m.f / is closed uder fiie uios, ad hece is a field. Proof. Halmos Mootoe Class Theorem is proved i every textbook. See Billigsley (995, Theorem 3.4), Ash ad Doléas-Dade (2000, Theorem.3.9), or Chug (200, Theorem 2..2), amog others. The above olie is similar to Chug (200). (a) By defiitio. I fact,.f / D m.f /. (b) By Exercise.39: A field is a -field iff it is also a M.C. If m.f / is a field, the it is a -field cotaiig F ; hece,.f / m.f /. (c) 2 F m.f /. (d) Let fa g G be mootoe; the A c is also mootoe. The DeMorga idetities 0 c 0 c [ \ \ A D A c ; A D A show that G is a M.C. Sice F is closed uder complemets ad F m.f /, it is clear that F G. Hece m.f / G by the miimality of m.f /. (e) By (d), m.f / G, which meas that for ay A 2 m.f /, we have A c 2 m.f /. Hece, m.f / is closed uder implemetatio. (f) Let G D A W A [ B 2 m.f / for all B 2 F. If fa g G is mootoe, the fa [ Bg is also mootoe. The idetities 0 0 [ [ \ A [ B D.A [ B/ ; A [ B D.A [ B/ A show that G is a M.C. Sice F is closed uder fiite uios ad F m.f /, it follows that F G, ad so m.f / G by the miimality of m.f /. (g) As i (f) we ca show G 2 is a M.C. By (f), m.f / G, which meas that for ay A 2 m.f / ad B 2 F we have A [ B 2 m.f /. This i tur meas that F G 2. Hece, m.f / G 2. A A A c