2.4 Measure Theory 21 From De Morgan’s laws it furthermore follows that A is also closed under countable intersections, i.e. if Ai ∈Afor i ∈Nthen ∞\ i=1 Ai ∈A. The pair (X, A) is called a measurable space, and elements of A are called measurable sets. Measure, measure space A measure µ on a measurable space (X, A) is a functionµ: A→[0, ∞] such that 1. µ(∅)=0, i.e. the empty set has measure 0; 2. if {Ai ∈A}i ∈N is a countable collection of pairwise disjoint sets inA, then µ ∞[ i=1 Ai!= ∞X i=1 µ(Ai). The triple (X, A,µ) is called a measure space. Note that the co-domain of µ includes ∞; a measurable set can have infinite measure. Probability measure, probability space Aprobability measure is a measure with total measure one, i.e. µ(X) = 1. In this context, a probability space is a measure space with a probability measure. Measurable function Let (X, A) and(Y, B) be measurable spaces. A function f : X→Y is measurable if f−1(B) ∈Afor all B∈B. Pushforward Let (X, A) and(Y, B) be measurable spaces, f : X→Y ameasurable function, andµ: A→[0, ∞] a measure on(X, A), then the pushforward of µis defined to be the measure f#µ: B→[0, ∞] given by f#µ(B)=µ(f−1(B)) for B∈B.
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