Prove $E(X|Y)=0$ given that $E[Xg(Y)]=0$ for any measurable function $g$.

Question

Please help me prove $E(X|Y)=0$ given that, for any measurable function $g$: $$E[Xg(Y)]=0$$

I have been trying using a definition of conditional expectation, but it does not seem to work.

Thanks!

Answer 1

Hint: there is a measurable function $h$ such as $E[X|Y] = h(Y)$. And you also have, for any well behaved $g$, $$ E[Xg(Y)] = E[E[X|Y] g(Y)] $$