Let $X,Y \in L^1$ and
$$\mathbb E(X \vert Y)=Y$$ $$\mathbb E(Y \vert X)=X.$$
Is it then true that $X=Y.$
This question came up because I can show it for $X,Y \in L^2$ by just computing $E((X-Y)^2)$ but I fail to see whether it also holds by just assuming integrability.