If $E(U|X)=0$, then $E(U)=0$?

Question

If $U$ and $X$ are random variables such that $E(U|X)=0$, then $E(U)=0$.

Really? how to prove?

Answer 1

$E(U\mid X)$ is actually a random variable which is a certain function of $X$ (say, $g(X)$ for instance). There's a property that says that if you take its expectation you get $$E\big(E(U\mid X)\big)=E(U).$$

So, $$E(U)=E\big(E(U\mid X)\big)=E(0)=0.$$