Vol. 79, No. 1, 1978

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Probability measures and the C-sets of Selivanovskij

Steven Eugene Shreve

Vol. 79 (1978), No. 1, 189–196

Let X be a Borel space and 𝒮(X) the smallest σ-field containing the Borel subsets of X and closed under operation (A). Then 𝒮(X) is a sub-σ-field of the class of universally measurable subsets of X. Let P(X) be the space of probability measures on the Borel subsets of X and equip P(X) with the weak topology. It is proved that if A ∈𝒮(X), then {p P(X) : P(A) λ} is in 𝒮[P(X)] for every real λ. This allows us to shows that if X and Y are Borel spaces, f is an 𝒮(X × Y ) measurable extended-real-valued function, and x q(x) is 𝒮(X) measurable from X to P(Y ), then x f(x,y)q(dyx) is 𝒮(X) measurable on the set where it is defined. (Measurability is relative to the Borel σ-fields in the range spaces.)

Mathematical Subject Classification 2000
Primary: 28A05
Secondary: 28A20
Received: 20 March 1978
Revised: 12 June 1978
Published: 1 November 1978
Steven Eugene Shreve