integrable random variables $X,Y$ such that $E(X|Y)=X$ a.s. and $E(Y|X)=Y$ a.s.

Question

Let $X,Y$ are random variables such that $E(|X|)+E(|Y|)<\infty$, and the random variable $E(X|Y)=X$ a.s. and $E(Y|X)=Y$ a.s .

Then is it true that $X=Y$ a.s. ?

Answer 1

If $Y=X^{3}$ then the two equations are satisfied but $X =Y$ may not hold.

Answer 2

Let $Y=2X$ for any nondegenerate and integrable RV $X.$