Chapter 0 Review of Set Theory

Chapter 0

Review of set theory

Set theory plays a central role in the theory of probability. Thus, we will open this course with a quick review of those notions 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 elements of X will often be called points. If A is a subset of X and x is a point in X, then the notation x A means that x belongs to A, i.e., it is one of the points in A. If x does not belong to ∈ A, we write x A. Whenever the identity of the whole space X is clear, we will refer to subsets of X ∈ simples as sets. If A and B are sets, the notations

A B or B A mean that A is a subset of B, i.e.,⊂ every point in ⊃A is also a point in B. Two sets are called equal if they contain exactly the same set of points, i.e.,

A B and B A.

⊂ ⊂ 6 Chapter 0

When considering the collection of all subsets of X, we always include the empty set, which is denoted by . By deﬁnition,

x X, x .

For every set A, it always holds that∀ ∈ ∈

A X.

We will often consider sets of sets. To⊂ avoid⊂ confusion, we will use the term a collection (42&!) of sets, rather then a set of sets; however remember, a collection is nothing but a set. Thus, is is a collection of sets and A is a set, then A means that the set A is one of the elements of the collection of sets . The notation A , on the other hand, is meaningless.C ∈ C C Example⊂ C : Let X be a whole space. The collection of all its subsets is denoted by 2X; it is called the power set of X (%8'(% ;7&"8). This notation is suggestive, as if X has ﬁnite cardinality X , then

2X 2 X . =

Deﬁnition 0.1 A sequence An of sets is called increasing (%-&3) if ▲▲▲

( )A1 A2 A3 .

It is called decreasing (;$9&*) if ⊂ ⊂ ⊂

A1 A2 A3 .

In either case it is called monotone.⊃ ⊃ ⊃

To denote sets, we will use curly brackets. For example, if x, y X, then

x, y ∈

n denotes the set whose only elements are{ x}and y. If xk k 1 is a ﬁnite sequence of distinct points in X, we will denote the set whose only elements are the elements = of that sequence by ( ) xk k 1,...,n .

{ ∶ = } Review of set theory 7

If xk is an inﬁnite sequence of distinct points in X, we will denote the set whose only elements are the elements of that sequence by ( )

xk k N .

{ ∶ ∈ } Suppose that for every x X, ⇡ x is a proposition assuming values either True or False. Then, ∈ ( ) x X ⇡ x

denotes the set of all points in X{for∈ which∶ ( the)} proposition ⇡ x is True. More generally, if ⇡1 x ,⇡2 x ,... is a sequence of propositions regarding the point x, then ( ) ( ) ( ) x X ⇡1 x ,⇡2 x ,...

denotes the set of all points{ in ∈X for∶ which( ) all( the) propositions} ⇡k x are True.

( ) Example: Let X N and for n N let

= ∈ True n is even ⇡ n False n is odd. ( ) = Then, n N ⇡ n

denotes the set of even integers. In{ many∈ ∶ instances,( )} we will use the more intuitive notation n N n is even .

{ ∈ ∶ } As a tautological, yet useful remark: every set A can be expressed using the curly ▲▲▲ bracket notation as A x X x A .

Finally, for every x X, x is a= subset{ ∈ of∶X, i.e.,∈ } an element of 2X; such a subset is called a singleton (0&$*(*). Be careful to distinguish x from x . ∈ { } { } 8 Chapter 0

0.2 Unions and intersections

Let be a collection of subsets of X. The subset of X including all those points that belong to at least one set of the collection is called the union ($&(*!) of the collection,C and it is denoted by C or A A .

In the case where the collection C includes{ only∶ two∈ C} sets, e.g., A, B , we simply write C A B. C = { } If C = ∪ Ai i 1,...,n is a ﬁnite collection of sets, weC = write{ ∶ = } n Ai. i 1 C = If = Ai i N is a countable collection of sets,C we= write{ ∶ ∈ }

Ai. ∞ i 1 C = Finally, if = A is a non-countable collection ofC sets,= { where∶ ∈is} an index set, we write

C = ∈ By convention if is an empty collection of sets, then

C .

It is easy to see that whenever A B, thenC =

⊂ A B B.

∪ = Review of set theory 9

In particular, it always holds that

A A and A X X.

Moreover, if An is an∪ increasing= sequence of sets,∪ then=

( ) n Ak An. k 1

= = Let be a collection of subsets of X. The subset of X including all those points that belong to all sets of the collection is called the intersection (+&;*() of the collection,C and it is denoted by C or A A .

In the case where the collection C includes{ only∶ two∈ C} sets, e.g., A, B , we simply write C A B. C = { }

If C = ∩ Ai i 1,...,n is a ﬁnite collection of sets, weC = write{ ∶ = } n Ai. i 1

C = = If Ai i N is a countable collection of sets,C we= write{ ∶ ∈ }

Ai. ∞ i 1

C = = Finally, if A is a non-countable collection ofC sets,= { where∶ ∈is} an index set, we write

C = ∈ 10 Chapter 0

By convention if is an empty collection of sets then

C X.

C = It is easy to see that whenever A B, then

⊂ A B A.

In particular, it always holds that ∩ =

A and A X A.

Moreover, if An is a decreasing∩ = sequence of sets,∩ then=

( ) n Ak An. k 1

= = Deﬁnition 0.2 Two sets A and B are called disjoint (.*9') if they have no points in common, i.e., A B .Adisjoint collection of sets is a collection of sets such that every two distinct sets in this collection are disjoint. ∩ = Since unions of disjoint collections of sets play an important role in the theory of probability, we will denote the union of a disjoint collection by . For example,

A B denotes the union of the disjoint sets A and B. Unions and intersections satisfy the following important properties:

1. Both are commutative and associative. 2. Intersections are distributive over unions,

A B C A B A C .

3. Unions are distributive∩ over( ∪ intersections,) = ( ∩ ) ∪ ( ∩ )

A B C A B A C .

∪ ( ∩ ) = ( ∪ ) ∩ ( ∪ ) Review of set theory 11

0.3 Complements

Let A be a set. It complement (.*-:/) is the set of points which do not belong to A, Ac x x A .

Complementation satisﬁes the following= { ∶ properties:∈ } 1. For all sets A, A Ac X and A Ac .

c c 2. For all sets A, A ∪A. = ∩ = 3. c X and Xc . ( ) = 4. If A B then Ac Bc. = = 5. For every (possibly non-countable) collection of sets, ⊂ ⊃ c A A Ac C A ,

and ({ ∶ ∈ C}) = { ∶ ∈ C} A A c Ac A .

Let A and B be sets. The( di↵{erence∶ between∈ C}) =Aand{ B∶is the∈ C set} of all those points that are in A but not in B, namely,

A B x x A and x B A Bc.

= { ∶ ∈ ∈ } = ∩ 0.4 Limits

Let An be a sequence of sets. The superior limit (0&*-3 -&"#) of this sequence is defined as the set of points that belong to infinitely many of those sets: ( ) lim sup An x for all k there exists an n k such that x An . n = { ∶ ≥ ∈ } If x lim supn An we say that x An infinitely often (%1:1 05&!"). The inferior limit (0&;(; -&"#) of this sequence is defined as the set of points that ∈ ∈ belong to all but finitely many of those sets:

lim inf An x there exists k such that x An for all n k . n = { ∶ ∈ ≥ } 12 Chapter 0

If x lim supn An we say that x An eventually ($*/; 9"$ -: &5&2"). Note that both superior and inferior limits always exist. Also, it always holds that ∈ ∈ lim inf An lim sup An. n n ⊂ In the event that the superior limit and the inferior limit of a sequence of sets coincide, we call this set the limit of the sequence, namely,

lim An lim sup An lim inf An. n n n = = By deﬁnition, for k N,

∈ An x there exists an n k such that x An , ∞ n k = { ∶ ≥ ∈ } hence, =

An lim sup An. ∞ ∞ k 1 n k n

Likewise, for k N, = = =

∈ An x x An for all n k , ∞ n k = { ∶ ∈ ≥ } hence, =

An lim inf An. ∞ ∞ n k 1 n k

= = = Proposition 0.1 If An is a monotone sequence of sets, then it has a limit. Specif- ically, if An is increasing then ( ) ( ) lim An An. n ∞ n 1

= = and if An is decreasing then

( ) lim An An. n ∞ n 1

= = Review of set theory 13

Proof : Let An be increasing. Then for every k N,

An An and Ak. ( ) ∞ ∞ ∈ ∞ n k n 1 n k

Thus, = = = =

lim sup An An An, ∞ ∞ ∞ n k 1 n 1 n 1 and = = = = = lim inf An An. n ∞ k 1

Let An be decreasing. Then for every k =N=,

An An and Ak. ( ) ∞ ∞ ∈ ∞ n k n 1 n k

Thus, = = = =

lim sup An Ak, ∞ n k 1 and = = lim inf An An An. n ∞ ∞ ∞ k 1 n 1 n 1 n = = = = = Example: Let X be the set of integers, N, and let 2, 4, 6,... n is even An 1, 3, 5,... n is odd. { } Then, = { } lim sup An X and lim inf An , n n i.e., the sequence An→∞does not= have a limit. →∞ = Example: Let again( X) N and let ▲▲▲ j = Ak k j 0, 1, 2,... , namely, = ∶ = A1 1 A2 1, 2, 4,... A2 1, 3, 9,... , etc. Then, = { } = { } = { } lim Ak 1 . k = { } ▲▲▲ 14 Chapter 0

0.5 Algebras and -algebras of sets

Deﬁnition 0.3 A collection of sets is called an algebra of sets (-: %9"#-! ;&7&"8) if it satifsies the following properties: C 1. If A then Ac . , 2. If A ∈BC the A∈ CB . 3. X . ∈ C ∪ ∈ C ∈ C Proposition 0.2 Let be an algebra of sets. Then,

1. . C 2. If A ,...,A then n A . ∈ 1C n i 1 i 3. If A ,...,A then n A . 1 n ∈ C ∪i=1 i ∈ C 4. Let A, B . Then their di↵erence A B is in . ∈ C ∩ = ∈ C ∈ C C

Proof : Since is closed under complementation and X ,

C Xc . ∈ C

The second assertion follows by induction. = ∈ C The third assertion follows from the second assertion and de Morgan’s Law,

n n c c Ai Ai . i 1 i 1

= = = ∈ C The fourth assertion holds because

A B A Bc.

= ∩ n Thus, an algebra of sets is a collection of sets closed under ﬁnitely many set- theoretic operations of union, intersection and complementation. Review of set theory 15

Example: Let A be a set. Then, the collection of sets

, A, Ac, X is an algebra of sets. C = { }

▲▲▲ Example: The collection of all subsets 2X is an algebra of sets.

▲▲▲ Example: Suppose that X is an infinite set. The collection of all finite subsets of X is not an algebra of sets. Neither is the collection of all infinite subsets of X.

Deﬁnition▲▲▲ 0.4 A collection of sets is called a -algebra of sets if it is an algebra of sets, and in addition, if An is a sequence of sets in , then C ( ) C An . ∞ n 1

= ∈ C

Proposition 0.3 Let be a algebra of sets. If An is a sequence of sets in , then C ( ) C An . ∞ n 1

= ∈ C

Proof : This is an immediate consequence of the identity

c c An A . ∞ ∞ n n 1 n 1 = = = n That is, a -algebra of sets is closed with respect to countably many set-theoretic operations.

Example: The collection of all subsets 2X is a -algebra of sets.

▲▲▲ 16 Chapter 0

Proposition 0.4 Let , be a (possible non-countable) family of -algebras of sets. Then, their intersection, C ∈ F is also a -algebra of sets. = {C ∶ ∈ }

Proof : Let An be a sequence of sets in F . By deﬁnition, An for every . Since is a -algebra, then ( ) ∈ C ∈

C c X , A and An . n ∞ n 1

∈ C ∈ C = ∈ C This this holds for every , then

∈ c X F , A F and An F . n ∞ n 1

∈ ∈ = ∈ n

Deﬁnition 0.5 Let be a non-empty collection of sets. The -algebra generated by is the intersection of all those -algebras containing . C C is a -algebra and C .

Since 2X is a -algebra(C) = containing{A ∶ A , this intersection isC not⊂ A} empty. By Proposi- tion 0.4, is a -algebra. C (C) Example: Let A be a set. Then,

A , A, Ac, X .

({ }) = { }

. Exercise 0.1 Let A, B be sets. Find A, B . ▲▲▲

({ }) Review of set theory 17

0.6 Inverse functions

In calculus, you learned that a function f X Y is invertible only if it is both one-to-one and onto; in this case, we can define ∶ → f 1 Y X. − An inverse function, however, can always∶ → be defined as a mapping between sets. For every function f X Y we can define

∶ → f 1 2Y 2X − by ∶ → f 1 A x X f x A , A Y. − An important property of( ) the= { inverse∈ ∶ function( ) ∈ f}1 is that⊂ it preserves (commutes with) set-theoretic operations: −

Proposition 0.5 Let f X Y. Then,

1. For every A Y ∶ → f 1 A c f 1 Ac . ⊂ − − 1 1 2. If A, B Y are disjoint so are( f ( ))A ,=f B( )X. 3. f 1 Y X. − − − ⊂ ( ) ( ) ⊂ 4. If An Y is a sequence of subsets, then ( ) =

⊂ 1 1 f An f An . ∞ ∞ − n 1 n 1 − = = = ( )

Proof : Just follow the deﬁnitions. For example,

x f 1 A i↵ f x A, − hence ∈ ( ) ( ) ∈

x f 1 A c i↵ f x A i↵ f x Ac i↵ x f 1 Ac . − − ∈ ( ( )) ( )∈ ( ) ∈ ∈ ( ) 18 Chapter 0

n The fact that inverse functions commute with set-theoretic operations has the following implication. Let X and Y be sets and let f X Y. Let F 2Y be a -algebra of subsets of Y, then ∶ → ⊂ f 1 A A F − is a -algebra of subsets of X. { ( ) ∶ ∈ }