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Let $X,Y$ be independent random variables on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ taking values in the measurable spaces $(E_1,\mathcal{E}_1),(E_2,\mathcal{E}_2)$. Consider a $\mathcal{E}_1\otimes\mathcal{E}_2$-measurable function $\phi:E_1\times E_2\to\mathbb{R}$ such that $\mathbb{E}[|\phi(X,Y)|]<\infty$ and for all $x\in E_1$ we have $\mathbb{E}[|\phi(x,Y)|]<\infty$.

How can I calculate $\mathbb{E}[\phi(X,Y)|\sigma(X)]$? Any ideas for me? Thanks.

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The result is $\psi(X)$, where $\psi(x)=\mathbb E(\phi(x,Y))$. Any luck with your other question? – Did Dec 3 '12 at 9:49
I've also read about the title "Conditional Expectation of function of two RVs, one measurable, one independent". The thing is I don't know what's the point of this exercise is. Can you tell me? – Ichigo Dec 3 '12 at 10:25
Sorry, but I fail to understand the point of your last comment. – Did Dec 3 '12 at 10:28
What is $\sigma(X)$? Where does this problem come from? – Potato Jan 11 at 19:22

closed as too localized by Did, TMM, tomasz, Quixotic, Davide Giraudo Jan 11 at 21:50

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