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I am given two integrable random variables $X$ and $Y$, where $\mathbb{E}(X\mid Y)=Y$ almost surely (a.s.) and $\mathbb{E}(Y\mid X)=X$ a.s. If the variances of $X$ and $Y$ exist, I can prove easily that $X=Y$ a.s. if I use the Cauchy-Schwarz-inequality. My question is, does this result also hold for integrable random variables? And if so, why?

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Complete solution deleted, due to OP's inappropriate behaviour in the comments. –  Did Nov 21 '12 at 9:50
Just to understand the things right: it is inappropriate to point out errors? –  johanMozart Nov 21 '12 at 9:55
Sure, this is exactly what I meant, of course... Only making your case worse. –  Did Nov 21 '12 at 10:08
I am speechless. –  johanMozart Nov 21 '12 at 10:12

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