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Let $\mathcal{A}\subseteq\mathcal{B}$ be a sub $\sigma$-algebra on the probability space $(\Omega,\mathcal{B},\mu)$, $X,Y$ real random variables being $X$ $\mathcal{B}$-measurable and $Y$ $\mathcal{A}$-measurable. It is true that $$ \mathbb{E}(XY|\mathcal{A}) = \mathbb{E}(X|\mathcal{A})Y? $$ I know it is true for bounded $Y$ and integrable $X$, but I think it also is true whenever $X$ and $XY$ are integrable (we only need $\mathbb{E}(X),\mathbb{E}(XY)$ to be defined)

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    $\begingroup$ Yes, it is true and it can be proved by approximating $X$ and $Y$ by bounded random variables. $X_n=X1_{|X|<n}$ and $Y_n=Y1_{|Y|<n}$ will work. $\endgroup$ – Georgii Riabov May 26 '18 at 18:31
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The mentioned statement is true. After having decomposed $X$ and $Y$ into positive and negative part, it suffices to treat the case where $X$ and $Y$ are both non-negative, by linearity of conditional expectation. Define $Y_n$ by giving the value $Y$ if $Y\leqslant n$ and zero otherwise. Then $Y_n$ is a bounded $\mathcal A$-measurable random variable hence $$ \tag{*}\mathbb{E}(XY_n|\mathcal{A}) = \mathbb{E}(X|\mathcal{A})Y_n $$ Since $Y_n\leqslant Y_{n+1}$ almost surely, $Y_n\to Y$ a.s. and $X$ is non-negative, the monotone convergence theorem for conditional expectation gives that the left hand side of (*) converges to $\mathbb{E}(XY|\mathcal{A})$ almost surely.

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Yes you are right. It is true whenever X and XY are integrable, i.e., their expectation is well-defined.

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    $\begingroup$ I think the Question here deserves more of a reasoned mathematical argument. As phrased you are not adding more information than what the Question itself contains, and less than what was provided in an earlier Comment. $\endgroup$ – hardmath May 26 '18 at 19:14
  • $\begingroup$ Ah ok, actually I am very new in Stack Exchange and I just wanted to clear the doubt in the question even before the previous comment as I knew it is always true, but couldn't provide a quick mathematics. Anyway thank you very much for your suggestion! $\endgroup$ – Anamitra Chaudhuri May 27 '18 at 3:14

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