Proof of $E[XY|\mathscr{H}]=XE[Y|\mathscr{H}]$ for $\mathscr{H}$-measurable $X$ I reckon this is a pretty trivial result in the context of elementary probability, but how would one show it in the context of measure-theoretic probability theory?
That is, that $E[XY|\mathscr{H}]=XE[Y|\mathscr{H}]$ for $\mathscr{H}$-measurable $X$ where $X$ and $Y$ are random variables.

When $X=I_A$ for some measurable set $A$ (w.r.t. $\mathcal H$) this follows from definition of conditional expectation. Now go to simple functions and then take limits. This gives the result for $X$ non-negative. Now write $X$ as $X^{+}-X^{-}$. You have to assume that $E|XY| <\infty$ for the result to be true. 
