Zero conditional mean and zero correlation

Question

$\newcommand{\Cov}{\mathrm{Cov}}$ $\newcommand{\E}{\mathrm{E}}$ Is $E(Y|X)=0$ equivalent to $\Cov(X,Y)=0$?

I know $E(Y|X)=0$ implies $\Cov(X,Y)=0$, because

$\Cov(X,Y) = \E(XY) - \E(X)\E(Y) = \E[\E(XY|X)]-E[X]E[E(Y|X)]=0$

But is the other way around true? Does $\Cov(X,Y)=0$ imply $E(Y|X)=0$?

Can anyone give a proof or disproof? Thanks.

Answer 1

No.

Let $Y=c$ a.s. with $c\neq0$.

Then $\mathsf{Cov}(X,Y)=0$ but $\mathsf E(Y\mid X)=c\neq0$.