Iterated expectations conditional on an event

Question

I know that $E(E(X|Y)) = E(X)$ from the law of iterated expectations but what if we let $B$ be a subset of the real line, then is it true that $E[E[X|Y]|Y \in B]=E[X|Y \in B]$? If so, how can I prove it?

Answer 1

First，we know $E(X|Y)$ is a $\sigma(Y)$ measurable function, so when $X$ is $\sigma$ measurable $E(X|Y)=X$.

Second,we have to use a property of conditional expectation:

If $\mathcal C\subset \mathcal C'\subset \mathcal F,$ then $E[E[X|\mathcal C]|\mathcal C']= E[X|\mathcal C] = E[E[X|\mathcal C']|\mathcal C]~~~~~~P_{\mathcal C}~~~a.e.$

By these conclusions,$E[E[X|Y]|Y\in B]=E[E[X|Y]|Y\mathbb I_B]=E[E[X|Y\mathbb I_B]|Y]=E[X|Y\mathbb I_B]=E[X|Y\in B]$.