Measurable sets of measures

if and only if the function co -»φ(ω)(A) is measurable from (Ω, Ssf) to the Borel line for each A e j^~.Proof.Routine.2.2.If φ is a measurable map from (Ω, Ssf) to {M, 2 1 *), and f is a bounded, measurable function from (Ω x X, J^f x Σ) to the Borel line, then ω -• I f(ω, x)φ{ω){dx) is a measurable function from (Ω, sf) to the Borel line.Proof.Extend from indicators of measurable rectangles. The σ-field Σ* is countably generated.