Please confirm this equality of conditional expectation?

Question

I know that we have $E[E[Y\mid X]]=E[Y]$

Given $I_A$ is the indicator function on set $A$, do we have the following equality and why?

$$E[E[Y\mid X]I_A]=E[YI_A]$$

proof or intuition or counterexample will be much appreciated.

Answer 1

The identity holds if $A$ belongs to the sigma-algebra $\sigma(X)$, otherwise it is not guaranteed. When $A$ is in $\sigma(X)$, the identity is one of the two requirements that $E(Y\mid X)$ must meet as the conditional expectation of $Y$ conditionally on $X$.

Three-points counterexample with $X$ equal on two points exactly, to make sure $\sigma(X)$ is neither $\{\varnothing,\Omega\}$ nor $2^\Omega$.