Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.
The hint I was given is to evaluate: $$E[X-Y;X>a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$
which I can write as: $$\int_A(X-Y)dP +\int_B(X-Y)dP$$ where $A=\{X>a, Y\leq a\}$ and $B=\{X\leq a,Y\leq a\}$.
But I need some more hints.