A Probability Path.pdf

# 1 complementation the complement of a subset a c q is

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1. Complementation : The complement of a subset A c Q is Ac : ={w:w¢A} . 2. Intersection over arbitrary index sets : Suppose T is some index set and for each t e T we are given A 1 c Q . We define n A, := {w: we A 1, V t e T}. reT The collection ofsubsets {A 1, t e T} is pairwise disjoint if whenever t, t' e T, butt :ft t', we have A 1 nA,, = 0. A synonym for pairwise disjoint is mutually disjoint. Notation : When we have a small number of subsets , perhaps two, we write for the intersection of subsets A and B AB =AnB , using a "multiplication" notation as shorthand. 3. Union over arbitrary index sets: As above, let T be an index set and suppose A 1 c Q . Define the union as UAr := {w: we A, , for some t e T} . lET When sets At. Az, .. . are mutually disjoint, we sometimes write At +Az + .. . or even L:f:t A; to indicate Uf:t A;, the union of mutually disjoint sets . 4. Set difference Given two sets A, B, the part that is in A but not in B is A \B : =ABc. This is most often used when B C A; that is, when AB =B . 5. Symmetric difference : If A, Bare two subsets, the points that are in one but not in both are called the symmetric difference A !::. B = (A \B) U (B \A).

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4 1. Sets and Events You may wonder why we are interested in arbitrary index sets. Sometimes the natural indexing of sets can be rather exotic. Here is one example . Consider the space USC+([O, oo)), the space of non-negative upper semi-continuous functions with domain [0, oo). For f e USC+([O, oo)), define the hypograph hypo(/) by hypo(/)= {(s,x): 0::: x::: f(s)}, so that hypo(/) is the portion of the plane between the horizontal axis and the graph of f. Thus we have a family of sets indexed by the upper semi-continuous functions, which is a somewhat more exotic index set than the usual subsets of the integers or real line. The previous list described common ways of constructing new sets from old. Now we list ways sets can be compared. Here are some simple relations between sets. 1. Containment: A is a subset of B, written A c B orB :J A, iff AB =A or equivalently iff w E A implies w e B. 2. Equality: Two subsets A, B are equal, written A = B, iff A c B and B c A. This means w e A iff w e B. Example 1.2.1 Here are two simple examples of set equality on the real line for you to verify. (i) nj(n + 1)) = [0, 1). (ii) (0, 1/n) = 0. 0 Here are some straightforward properties of set containment that are easy to verify: A cA, A c B and B c C implies A c C, A C C and B C C implies A U B c C, A :J C and B :J C implies AB :J C, A C B iff Be C A c Here is a list of simple connections between the set operations: 1. Complementation: 2. Commutativity of set union and intersection: AUB=BUA, AnB=BnA .
1.2 Basic Set Theory 5 Note as a consequence of the definitions, we have AUA =A, AU0=A, AUQ=Q, A UAC = Q , AnA= A, An0=0 An!'2=A , A nAc = 0. 3. Associativity of union and intersection: (A U B) U C = A U (B U C), (A n B) n C = A n (B n C). 4. De Morgan's laws, a relation between union, intersection and complemen- tation: Suppose as usual that T is an index set and A 1 c Q .

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