# Joint Expectation of independent Random Variables given two sigma-algebras

We have a question regarding two random variables $X$,$Y$ on a probability space with sigma-algebra $\mathcal{F}$ and a sub-sigma algebra $\mathcal{M}$ such that $X$ is independent of $\mathcal{M}$ and $Y$ is $\mathcal{M}$-measurable.

We are asked to show that $E(I(X+Y \text{ in } A)|\mathcal{M}) = E(I(X+Y \text{ in } A)|Y)$ a.s. where $I$ is the indicator function and $A$ is a Borel set on $\mathbb{R}$.

My attempt: Write $\mathcal{H} = \sigma(Y)$, the sigma algebra generated by $Y$. We have that $\mathcal{H}$ is a subset of $\mathcal{M}$, as $Y$ is $\mathcal{M}$-measurable.

Write $Z=I(X+Y \text{ in } A)$

Now, I want to show that $E[Z|\mathcal{H}]$ is a conditional expectation on $\mathcal{M}$, because then by the almost sure uniqueness of conditional expectation the result follows, and so we have to show firstly that it is a $\mathcal{M}$-measurable function, which follows as it is a $\mathcal{H}$-measurable function, and secondly that for any $A$ in $\mathcal{M}$, $E[E[Z|\mathcal{H}]I(\mathcal{M})]=E[Z I(\mathcal{M})]$.

I can't for the life of me seem to figure out this step, any hints would be appreciated.