If a statistic T is sufficient for P, then for every function f which is (A, P)-integrable

If a statistic T is sufficient for P, then for every function f which is (A, Pθ)-integrable for all θ ∈ there exists a determination of the conditional expectation function Eθ[ f (X) | t] that is independent of θ. [If X is Euclidean, this follows from Theorems 2.5.2 and 2.6.1. In general, if f is nonnegative there exists a nondecreasing sequence of simple nonnegative functions fn tending to f . Since the conditional expectation of a simple function can be taken to be independent of θ by Lemma 2.4.1(i), the desired result follows from Lemma 2.4.1(iv).]