Conditional expectation w.r.t measure and push-forward measure

Question

I have been introduce to the theory of conditional expectation. My question is simple. I wish to know the similarities between two quantities. My book talks about the existence of $E(Y|X=x)$ and $E(Y|\mathcal{G})$ as conditional expectations being $\mathcal{G}-measurable$ and verifying : \begin{align} \int_A YdP &= \int_A E(Y|\mathcal{G})dP \\ \int_{\{X\in B\}} YdP &= \int_{\{X\in B\}} E(Y|X=x)dP_X \end{align} I know how to go from left to right mainly by the variable change theorem and considering $X$ as the identity map, but in case of starting with X not being the identity and considering $\sigma(X)$ as the sub $\sigma-algebra$,how do we effectively come back to the general definition in terms of $\mathcal{G}$ ?

Answer 1

The relationship between $\mathrm{E}[Y\mid\mathcal{G}]$ with $\mathcal{G}=\sigma(X)$ (which is usually just written $\mathrm{E}[Y\mid X]$) and $\mathrm{E}[Y\mid X=x]$ is the following. If $\varphi(x)=\mathrm{E}[Y\mid X=x]$ for every $x$, then $\mathrm{E}[Y\mid X]=\varphi(X)$.

Now, for any set $A\in \sigma(X)$, there is a Borel set $B\subseteq\mathbb{R}$ such that $A=\{X\in B\}$. In particular, $\mathbf{1}_A=\mathbf{1}_B(X)$ and hence

$$ \int_A Y\,\mathrm dP=\int_A \mathrm{E}[Y\mid X]\,\mathrm dP=\int_\Omega \mathbf{1}_B(X)\varphi(X)\,\mathrm dP=\int_\mathbb{R} \mathbf{1}_B(x)\varphi(x)\,P_X(\mathrm dx), $$ where the first equality is by definition of the conditional expectation and the last equality is the change of variables theorem.