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(dy∣x) is 𝒮(X) measurable on the set where
it is defined. (Measurability is relative to the Borel σ-fields in the range
spaces.)
