Definition
Let $G$ be a sub $\sigma-$field of sets in a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$. For any $E \in \mathcal{F}$ there exists a unique $G$-measurable RV $\mathbb{P}(E|G):\Omega \rightarrow [0,1]$ that satisfies $\mathbb{P}(E \cap A) = \mathbb{E}[\mathbb{P}(E|G)*\textbf{I}_{A}]$
Questions
There are a couple of things about this definition that confuses me.
First, I'm not quite sure how a probability measure can defined as a random variable. I understand that a RV is a function $X(\omega)$: $\Omega \rightarrow R$. And technically they are defining $\mathbb{P}(E|G):\Omega \rightarrow [0,1]$. However, where does $(\omega)$ act as an input? Is it $\mathbb{P}(E|G(\omega))?$
Second, I'm trying to prove this equality (so that it makes more sense to me) using first principles.
$\mathbb{E}[\mathbb{P}(E|G)*\textbf{I}_{A}]$ = $\int_{\Omega} \textbf{I}_{A}\mathbb{P}(E|G)d\mathbb{P}$ (I think) and I'm not quite sure what an appropriate next step would be so that we eventually get $\mathbb{P}(E \cap A)$.
Any help would be greatly appreciated! Thanks.