We have just introduced the notion of conditional expectations for sub-$\sigma$-algebras. Now let $(\Omega, \mathcal{F}, P)$ be a probability space and $\bigcup\limits_{n=1}^{N}A_{n}=\Omega$ be a partition. Define $\mathcal{A}:=\sigma(\{A_{n}:n=1,...,N \})$ as the sub-$\sigma$-algebra.
I know that $\mathbb E[X\vert \mathcal{A}]=\sum\limits_{n=1}^{N}\mathbb E[X\mid A_{n}]1_{A_{n}}$
But in a textbook, it is stated -without explanation- that $\mathbb E[X\mid \mathcal{A}]$ is $\mathcal{A}$-measurable.
I do not understand what is meant by this. I mean are we talking about measurability in terms of functions or of sets? I am confused.