Proving that sum of two measurable functions is measurable for conditional expectation

Question

I'm trying to show something that seems pretty simple: $\mathbb{E}[aX + Y | \mathcal{G}] = a\mathbb{E}[X | \mathcal{G}] + \mathbb{E}[Y | \mathcal{G}]$ where the conditional expectation is defined such that it is measurable with respect to $\mathcal{G}$ and $\mathbb{E}[1_A X] = \mathbb{E}[1_A \mathbb{E}[X | \mathcal{G}]]$ for all $A \in \mathcal{G}$. However, I can't seem to figure out why $a\mathbb{E}[X | \mathcal{G}] + \mathbb{E}[Y | \mathcal{G}]$ needs to be measurable. Why is this necessarily the case?

Answer 1

The set of $\mathcal{G}$-measurable functions is closed under addition and multiplication. Since $\mathbb{E}[X\mid\mathcal{G}]$, $\mathbb{E}[Y\mid\mathcal{G}]$ and $a$ are all $\mathcal{G}$-measurable, it follows that $a\mathbb{E}[X\mid\mathcal{G}]+\mathbb{E}[Y\mid\mathcal{G}]$ is also $\mathcal{G}$-measurable.