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Suppose $X$ and $Y$ are two random variables defined on a probability space $(\Omega, \mathcal{F},P)$ with joint density function $f_{X,Y}$. For any $B \in \mathcal{B}(\mathbb{R})$, why is it that $ \int_\mathbb{R} y \left( \int_\mathbb{R} I_B(x) f_{X,Y}(x,y) \,dx \right) \, dy = \int_\Omega Y I_{X^{-1}(B)} \, dP \quad ? $ Note: This is one step needed in my attempt to prove that the elementary definition and theorectical definition of $E(Y|X)$ agree a.e. after applying Fubini's Theorem.

Thanks!

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    @Byron$S$chmuland: I now understand it. Thanks!2011-11-08

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Because $\displaystyle\iint_{\mathbb R^2} \varphi(x,y)f_{X,Y}(x,y)\mathrm dx\mathrm dy=E(\varphi(X,Y))=\int_\Omega\varphi(X,Y)\mathrm d\mathrm P$ for the function $\varphi$ defined on $\mathbb R^2$ by $\varphi(x,y)=y\cdot[x\in B]$, as soon as $Y$ is integrable.