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 = Aγ γ Γ Γ is a non-countable collection ofC sets,= { where∶ ∈is} an index set, we write Aγ. γ Γ 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 Aγ γ Γ is a non-countable collection ofC sets,= { where∶ Γ∈is} an index set, we write Aγ. γ Γ 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.