1

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.

  • I'm fairly sure I can write a short proof that this is true if I put some time into it, but so far I'm coming up with a long proof. – Michael Hardy Jan 14 '18 at 00:28
  • 2
    . . . . . and now I find I answered this six years ago: https://math.stackexchange.com/questions/74692/conditional-expectation-and-almost-sure-equality – Michael Hardy Jan 14 '18 at 00:29

0 Answers0