Conditional expectation

[yavanna]

I don't know how to prove this one. Could you give me some hint? Thanks

Let X,Y r.v., $\displaystyle \mathbb{E}(X|Y)=Y$ and $\displaystyle \mathbb{E}(Y|X)=X$.
Proove that $\displaystyle X=Y$ a.s.

 

[tkhunny]

1) Your notation can't be right, can it? Don't you mean E(X|Y) = y and finally x = y?
2) What is the algebraic definition of your conditional expectations?