Suppose that two random variables X and Y are independent and we have

g(X,Y)=f(X)h(Y)g(X,Y)=f(X)h(Y).

Show that E(g(X,Y)=E(f(x))E(h(y))

Notice that f(x) here is just a function of x (not a density), and h(y) is another function but of Y. Could be a linear function, for example, but not necessarily.