If E[Y | X = x]=x how do I show that Y=X?

Question

I know that E[X|X]=X but I don't know how to bring this around to the problem.

Answer 1

The claim is false. Counter-example: $X = $ any r.v. and $Y = X + Z$ for some zero-mean r.v. $Z$ which is independent of $X$. Then

$$E[Y \mid X=x] = E[X + Z \mid X=x] = E[X \mid X=x] + E[Z \mid X=x] = x + E[Z] = x$$

but $Y \neq X$.