Chapter 0

Review of set theory

Set theory plays a central role in the theory of probability. Thus, we will openthis course with a quick review of those notions of set theory which will be usedrepeatedly.

0.1 Sets

Throughout this chapter, let X be a set, which we will call the whole space. Theelements of X will often be called points. If A is a subset of X and x is a point inX, 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 toA, we write

x �∈ A.

Whenever the identity of the whole space X is clear, we will refer to subsets of Xsimples 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 setsare 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 emptyset, 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 acollection (�42&!) of sets, rather then a set of sets; however remember, a collectionis nothing but a set. Thus, is C is a collection of sets and A is a set, then A ∈ Cmeans that the set A is one of the elements of the collection of sets C. The notationA ⊂ C, on the other hand, is meaningless.

Example: Let X be a whole space. The collection of all its subsets is denoted by2X; it is called the power set of X (�%8'(% ;7&"8). This notation is suggestive, as ifX 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}denotes the set whose only elements are x and y. If (xk)nk=1 is a finite sequence ofdistinct points in X, we will denote the set whose only elements are the elementsof 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 whoseonly 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 orFalse. Then,

{x ∈ X ∶ ⇡(x)}denotes the set of all points in X for which the proposition ⇡(x) is True. Moregenerally, 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

⇡(n) = �������True n is evenFalse n is odd.

Then,{n ∈ N ∶ ⇡(n)}

denotes the set of even integers. In many instances, we will use the more intuitivenotation

{n ∈ N ∶ n is even}.▲▲▲

As a tautological, yet useful remark: every set A can be expressed using the curlybracket 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 subsetis called a singleton (�0&$*(*). Be careful to distinguish x from {x}.

8 Chapter 0

0.2 Unions and intersections

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

�C or �{A ∶ A ∈ C}.In the case where the collection C includes only two sets, e.g., C = {A,B}, wesimply write �C = A ∪ B.

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

�C = n�i=1

Ai.

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

�C = ∞�i=1

Ai.

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

�C =��∈�A�.

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

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

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�k=1

Ak = An.

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

�C or �{A ∶ A ∈ C}.In the case where the collection C includes only two sets, e.g., C = {A,B}, wesimply write �C = A ∩ B.

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

�C = n�i=1

Ai.

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

�C = ∞�i=1

Ai.

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

�C =��∈�A�.

10 Chapter 0

By convention if C is an empty collection of sets then

�C = X.

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�k=1

Ak = An.

Definition 0.2 Two sets A and B are called disjoint (�.*9') if they have no pointsin common, i.e., A∩ B = �. A disjoint collection of sets is a collection of sets suchthat every two distinct sets in this collection are disjoint.

Since unions of disjoint collections of sets play an important role in the theory ofprobability, 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 toA,

Ac = {x ∶ x �∈ A}.Complementation satisfies the following properties:

1. For all sets A,A ∪ Ac = X and A ∩ Ac = �.

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

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

Let A and B be sets. The di↵erence between A and B is the set of all those pointsthat 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 isdefined as the set of points that belong to infinitely many of those sets:

lim supn

An = {x ∶ for all k there exists an n ≥ k such that x ∈ An}.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 thatbelong to all but finitely many of those sets:

lim infn

An = {x ∶ there exists k such that x ∈ An for all n ≥ k}.

12 Chapter 0

If x ∈ lim supn An we say that x ∈ An eventually (�$*/; 9"$ -: &5&2"). Note thatboth superior and inferior limits always exist. Also, it always holds that

lim infn

An ⊂ lim supn

An.

In the event that the superior limit and the inferior limit of a sequence of setscoincide, we call this set the limit of the sequence, namely,

limn

An = lim supn

An = lim infn

An.

By definition, for k ∈ N,

∞�n=k

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

k=1

∞�n=k

An = lim supn

An.

Likewise, for k ∈ N,

∞�n=k

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

k=1

∞�n=k

An = lim infn

An.

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

limn

An = ∞�n=1

An.

and if (An) is decreasing then

limn

An = ∞�n=1

An.

Review of set theory 13

Proof : Let (An) be increasing. Then for every k ∈ N,∞�

n=kAn = ∞�

n=1An and

∞�n=k

Ak.

Thus,lim sup

nAn = ∞�

k=1

∞�n=1

An = ∞�n=1

An,

andlim inf

nAn = ∞�

k=1An.

Let (An) be decreasing. Then for every k ∈ N,∞�

n=kAn = ∞�

n=1An and

∞�n=k

Ak.

Thus,lim sup

nAn = ∞�

k=1Ak,

andlim inf

nAn = ∞�

k=1

∞�n=1

An = ∞�n=1

An.

n

Example: Let X be the set of integers, N, and let

An =�������{2,4,6, . . .} n is even{1,3,5, . . .} n is odd.

Then,lim sup

n→∞ An = X and lim infn→∞ An = �,

i.e., the sequence (An) does not have a limit. ▲▲▲Example: Let again X = N and let

Ak = �k j ∶ j = 0,1,2, . . .� ,namely,

A1 = {1} A2 = {1,2,4, . . .} A2 = {1,3,9, . . .},etc. Then,

limk

Ak = {1}.▲▲▲

14 Chapter 0

0.5 Algebras and �-algebras of sets

Definition 0.3 A collection C of sets is called an algebra of sets (-: %9"#-!�;&7&"8) if it satifsies the following properties:

1. If A ∈ C then Ac ∈ C.

2. If A,B ∈ C the A ∪ B ∈ C.

3. X ∈ C.

Proposition 0.2 Let C be an algebra of sets. Then,

1. � ∈ C.2. If A1, . . . ,An ∈ C then ∪n

i=1Ai ∈ C.3. If A1, . . . ,An ∈ C then ∩n

i=1Ai ∈ C.4. Let A,B ∈ C. Then their di↵erence A � B is in C.

Proof : Since C is closed under complementation and X ∈ C,

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

n�i=1

Ai = � n�i=1

Aci�

c ∈ 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

C = {�,A,Ac,X}is an algebra of sets. ▲▲▲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 ofX is not an algebra of sets. Neither is the collection of all infinite subsets of X.▲▲▲Definition 0.4 A collection C of sets is called a �-algebra of sets if it is an alge-bra of sets, and in addition, if (An) is a sequence of sets in C, then

∞�n=1

An ∈ C.

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

n=1An ∈ C.

Proof : This is an immediate consequence of the identity

∞�n=1

An = �∞�n=1

Acn�

c

.

n

That is, a �-algebra of sets is closed with respect to countably many set-theoreticoperations.

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

16 Chapter 0

Proposition 0.4 Let C�, � ∈ � be a (possible non-countable) family of �-algebrasof sets. Then, their intersection,

F =�{C� ∶ � ∈ �}is also a �-algebra of sets.

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

X ∈ C�, Acn ∈ C� and

∞�n=1

An ∈ C�.This this holds for every � ∈ �, then

X ∈F , Acn ∈F and

∞�n=1

An ∈F .n

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

�(C) =�{A ∶ A is a �-algebra and C ⊂ A}.Since 2X is a �-algebra containing C, this intersection is not empty. By Proposi-tion 0.4, �(C) is a �-algebra.

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 bothone-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

byf −1(A) = {x ∈ X ∶ f (x) ∈ A}, A ⊂ Y.

An important property of the inverse function f −1 is that it preserves (commuteswith) set-theoretic operations:

Proposition 0.5 Let f ∶ X → Y. Then,

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

f −1 �∞�n=1

An� = ∞�n=1

f −1(An).

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 fol-lowing 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.