Basic Question on Conditional Expectation being in $L^2$

Question

Suppose that $X,Y$ is a random-variables in $L^2(\Omega,\mathcal{F};\mathbb{P})$, where $(\Omega,\mathcal{F};\mathbb{P})$ is a complete probability space. Then the conditional expectation $E[X|\sigma(Y)]$ can be represented as a measurable function $$ g(Y)=E[X|\sigma(Y)], $$ see page 83.

This may be a silly question, but can the map $$ y\mapsto E[X|\sigma(Y^{-1}[y])], $$ be given meaning (or a modification of it), if so is it also in $L^2(\Omega,\mathcal{F};\mathbb{P})$ (or at-least in $L^2(\Omega,\mathcal{F};\mathbb{P})$ for some $p \in [1,\infty)$).

Answer 1

Fix a measurable version of $g$ and define $E(X|Y=y)$ to be $g(y)$. Note that $g(Y) \in L^{2}$.