.1. Let X,Y E L'(12, F, P) be independent random variables and let o(X) = G C F be the sub-o-algebra generated by X. Let f: R2 + R be a measurable function such that f(X,Y) E L'(12, F,P). Define the function g: R → R by g(x) = E[f(x,Y)] for each x E R. Prove that > E[F(X, Y)|0(X)] = g(X) a.s. (Hint: Use the Monotone Class Theorem.]
