Chapter 0. Review of set theory. 0.1 Sets

Transcription

1 Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly. 0.1 Sets Throughout this chapter, let X be a set, which we will call the whole space. The elemets of X will ofte be called poits. If A is a subset of X ad x is a poit i X, the the otatio x A meas that x belogs to A, i.e., it is oe of the poits i A. If x does ot belog to A, we write x A. Wheever the idetity of the whole space X is clear, we will refer to subsets of X simples as sets. If A ad B are sets, the otatios A B or B A mea that A is a subset of B, i.e., every poit i A is also a poit i B. Two sets are called equal if they cotai exactly the same set of poits, i.e., A B ad B A.

2 6 Chapter 0 Whe cosiderig the collectio of all subsets of X, we always iclude the empty set, which is deoted by. By defiitio, For every set A, it always holds that x X, x. A X. We will ofte cosider sets of sets. To avoid cofusio, we will use the term a collectio (42&!) of sets, rather the a set of sets; however remember, a collectio is othig but a set. Thus, is C is a collectio of sets ad A is a set, the A C meas that the set A is oe of the elemets of the collectio of sets C. The otatio A C, o the other had, is meaigless. Example: Let X be a whole space. The collectio of all its subsets is deoted by 2 X ; it is called the power set of X (%8'(% ;7&"8). This otatio is suggestive, as if X has fiite cardiality X, the 2 X = 2 X. Defiitio 0.1 A sequece (A ) of sets is called icreasig (%-&3) if It is called decreasig (;$9&*) if I either case it is called mootoe. A 1 A 2 A 3. A 1 A 2 A 3. To deote sets, we will use curly brackets. For example, if x, y X, the {x, y} deotes the set whose oly elemets are x ad y. If (x k ) k=1 is a fiite sequece of distict poits i X, we will deote the set whose oly elemets are the elemets of that sequece by {x k k = 1,...,}.

3 Review of set theory 7 If (x k ) is a ifiite sequece of distict poits i X, we will deote the set whose oly elemets are the elemets of that sequece by {x k k N}. Suppose that for every x X, (x) is a propositio assumig values either True or False. The, {x X (x)} deotes the set of all poits i X for which the propositio (x) is True. More geerally, if 1 (x), 2 (x),... is a sequece of propositios regardig the poit x, the {x X 1 (x), 2 (x),...} deotes the set of all poits i X for which all the propositios k (x) are True. Example: Let X = N ad for N let True () = False is eve is odd. The, { N ()} deotes the set of eve itegers. I may istaces, we will use the more ituitive otatio { N is eve}. As a tautological, yet useful remark: every set A ca be expressed usig the curly bracket otatio as A = {x X x A}. Fially, for every x X, {x} is a subset of X, i.e., a elemet of 2 X ; such a subset is called a sigleto (0&$*(*). Be careful to distiguish x from {x}.

4 8 Chapter Uios ad itersectios Let C be a collectio of subsets of X. The subset of X icludig all those poits that belog to at least oe set of the collectio C is called the uio ($&(*!) of the collectio, ad it is deoted by C or {A A C}. I the case where the collectio C icludes oly two sets, e.g., C = {A, B}, we simply write C = A B. If C = {A i is a fiite collectio of sets, we write If i = 1,...,} C = A i. i=1 C = {A i i N} is a coutable collectio of sets, we write Fially, if C = A i. i=1 C = {A } is a o-coutable collectio of sets, where C = A. By covetio if C is a empty collectio of sets, the C =. It is easy to see that wheever A B, the A B = B. is a idex set, we write

5 Review of set theory 9 I particular, it always holds that A = A ad A X = X. Moreover, if (A ) is a icreasig sequece of sets, the k=1 A k = A. Let C be a collectio of subsets of X. The subset of X icludig all those poits that belog to all sets of the collectio C is called the itersectio (+&;*() of the collectio, ad it is deoted by C or {A A C}. I the case where the collectio C icludes oly two sets, e.g., C = {A, B}, we simply write C = A B. If C = {A i is a fiite collectio of sets, we write If i = 1,...,} C = A i. i=1 C = {A i i N} is a coutable collectio of sets, we write Fially, if C = A i. i=1 C = {A } is a o-coutable collectio of sets, where C = A. is a idex set, we write

6 10 Chapter 0 By covetio if C is a empty collectio of sets the C = X. It is easy to see that wheever A B, the I particular, it always holds that A B = A. A = ad A X = A. Moreover, if (A ) is a decreasig sequece of sets, the k=1 A k = A. Defiitio 0.2 Two sets A ad B are called disjoit (.*9') if they have o poits i commo, i.e., A B =. Adisjoit collectio of sets is a collectio of sets such that every two distict sets i this collectio are disjoit. Sice uios of disjoit collectios of sets play a importat role i the theory of probability, we will deote the uio of a disjoit collectio by. For example, A B deotes the uio of the disjoit sets A ad B. Uios ad itersectios satisfy the followig importat properties: 1. Both are commutative ad associative. 2. Itersectios are distributive over uios, A (B C) = (A B) (A C). 3. Uios are distributive over itersectios, A (B C) = (A B) (A C).

7 Review of set theory Complemets Let A be a set. It complemet (.*-:/) is the set of poits which do ot belog to A, A c = {x x A}. Complemetatio satisfies the followig properties: 1. For all sets A, A A c = X ad A A c =. 2. For all sets A, (A c ) c = A. 3. c = X ad X c =. 4. If A B the A c B c. 5. For every (possibly o-coutable) collectio C of sets, ad ({A A C}) c = {A c ({A A C}) c = {A c A C}, A C}. Let A ad B be sets. The di erece betwee A ad B is the set of all those poits that are i A but ot i B, amely, A B = {x x A ad x B} = A B c. 0.4 Limits Let (A ) be a sequece of sets. The superior limit (0&*-3 -&"#) of this sequece is defied as the set of poits that belog to ifiitely may of those sets: lim sup A = {x for all k there exists a k such that x A }. If x lim sup A we say that x A ifiitely ofte (%1:1 05&!"). The iferior limit (0&;(; -&"#) of this sequece is defied as the set of poits that belog to all but fiitely may of those sets: lim if A = {x there exists k such that x A for all k}.

8 12 Chapter 0 If x lim sup A we say that x A evetually ($*/; 9"$ -: &5&2"). Note that both superior ad iferior limits always exist. Also, it always holds that lim if A lim sup A. I the evet that the superior limit ad the iferior limit of a sequece of sets coicide, we call this set the limit of the sequece, amely, lim A = lim sup A = lim if A. By defiitio, for k N, =k A = {x there exists a k such that x A }, hece, Likewise, for k N, k=1 =k A = lim sup A. =k A = {x x A for all k}, hece, k=1 =k A = lim if A. Propositio 0.1 If (A ) is a mootoe sequece of sets, the it has a limit. Specifically, if (A ) is icreasig the lim A = A. =1 ad if (A ) is decreasig the lim A = A. =1

9 Review of set theory 13 Proof : Let (A ) be icreasig. The for every k N, Thus, ad =k A = A =1 lim sup ad A = lim if k=1 =1 =k A = A, =1 A = A. k=1 Let (A ) be decreasig. The for every k N, Thus, ad =k A = A =1 lim if lim sup A = ad A = A k, k=1 k=1 =1 =k A = A. =1 A k. A k. Example: Let X be the set of itegers, N, ad let {2, 4, 6,...} is eve A = {1, 3, 5,...} is odd. The, lim sup A = X ad lim if A =, i.e., the sequece (A ) does ot have a limit. Example: Let agai X = N ad let amely, etc. The, A k = k j j = 0, 1, 2,..., A 1 = {1} A 2 = {1, 2, 4,...} A 2 = {1, 3, 9,...}, lim k A k = {1}.

10 14 Chapter Algebras ad -algebras of sets Defiitio 0.3 A collectio C of sets is called a algebra of sets (-: %9"#-! ;&7&"8) if it satifsies the followig properties: 1. If A C the A c C. 2. If A, B C the A B C. 3. X C. Propositio 0.2 Let C be a algebra of sets. The, 1. C. 2. If A 1,...,A C the i=1 A i C. 3. If A 1,...,A C the i=1 A i C. 4. Let A, B C. The their di erece A B is i C. Proof : Sice C is closed uder complemetatio ad X C, = X c C. The secod assertio follows by iductio. The third assertio follows from the secod assertio ad de Morga s Law, i=1 The fourth assertio holds because A i = c A c i C. i=1 A B = A B c. Thus, a algebra of sets is a collectio of sets closed uder fiitely may settheoretic operatios of uio, itersectio ad complemetatio.

11 Review of set theory 15 Example: Let A be a set. The, the collectio of sets C = {, A, A c, X} is a algebra of sets. Example: The collectio of all subsets 2 X is a algebra of sets. Example: Suppose that X is a ifiite set. The collectio of all fiite subsets of X is ot a algebra of sets. Neither is the collectio of all ifiite subsets of X. Defiitio 0.4 A collectio C of sets is called a -algebra of sets if it is a algebra of sets, ad i additio, if (A ) is a sequece of sets i C, the A C. =1 Propositio 0.3 Let C be a algebra of sets. If (A ) is a sequece of sets i C, the =1 A C. Proof : This is a immediate cosequece of the idetity A = c A c. =1 =1 That is, a -algebra of sets is closed with respect to coutably may set-theoretic operatios. Example: The collectio of all subsets 2 X is a -algebra of sets.

12 16 Chapter 0 Propositio 0.4 Let C, be a (possible o-coutable) family of -algebras of sets. The, their itersectio, F = {C } is also a -algebra of sets. Proof : Let (A ) be a sequece of sets i F. By defiitio, A C for every. Sice C is a -algebra, the X C, A c C ad A C. =1 This this holds for every, the X F, A c F ad A F. =1 Defiitio 0.5 Let C be a o-empty collectio of sets. The by C is the itersectio of all those -algebras cotaiig C. -algebra geerated (C) = {A A is a -algebra ad C A}. Sice 2 X is a -algebra cotaiig C, this itersectio is ot empty. By Propositio 0.4, (C) is a -algebra. Example: Let A be a set. The, ({A}) = {, A, A c, X}.. Exercise 0.1 Let A, B be sets. Fid ({A, B}).

13 0.6 Iverse fuctios Review of set theory 17 I calculus, you leared that a fuctio f X Y is ivertible oly if it is both oe-to-oe ad oto; i this case, we ca defie f 1 Y X. A iverse fuctio, however, ca always be defied as a mappig betwee sets. For every fuctio f X Y we ca defie f 1 2 Y 2 X by f 1 (A) = {x X f (x) A}, A Y. A importat property of the iverse fuctio f 1 is that it preserves (commutes with) set-theoretic operatios: Propositio 0.5 Let f X Y. The, 1. For every A Y ( f 1 (A)) c = f 1 (A c ). 2. If A, B Y are disjoit so are f 1 (A), f 1 (B) X. 3. f 1 (Y) = X. 4. If A Y is a sequece of subsets, the f 1 A = f 1 (A ). =1 =1 Proof : Just follow the defiitios. For example, x f 1 (A) i f (x) A, hece x ( f 1 (A)) c i f (x) A i f (x) A c i x f 1 (A c ).

14 18 Chapter 0 The fact that iverse fuctios commute with set-theoretic operatios has the followig implicatio. Let X ad Y be sets ad let f X Y. Let F 2 Y be a -algebra of subsets of Y, the is a -algebra of subsets of X. { f 1 (A) A F }