How to show the following about expectations

Question

If $X$ is $\mathcal{M}_1$ measurable on a probability space, $\mathcal{M}_2\subseteq\mathcal{M}_1$ and $Y$ is $\mathcal{M}_2$ measurable and they satisfy $E(XI_A)=E(YI_A)$ for every $A\in\mathcal{M}_2$, how would I be able to show that $E(XZ)=E(YZ)$ for every $Z$, $\mathcal{M}_2$ measurable? I have tried to approximate $Z$ by events of the type $I_{A_n}$ but for some reason cant seem to get it work.

Answer 1

Hints:

1. The condition $\mathbb{E}(X 1_A) = \mathbb{E}(Y 1_A)$ for all $A \in \mathcal{M}_2$ implies $Y = \mathbb{E}(X \mid \mathcal{M}_2)$.
2. Use tower property: $$\mathbb{E}(X Z) = \mathbb{E} \big( \mathbb{E}(X Z \mid \mathcal{M}_2) \big) = \dots$$

Remark: Note that we need some integrability assumption on $Z$; otherwise $\mathbb{E}(XZ)$ might not even be well-defined.