Let $X_1$ and $X_2$ be two random variables on $(\Omega,\mathcal{B},P)$. Suppose there is a function $g:\mathcal{B}\times\mathbb{R}\rightarrow[0,1]$ such that for any $x$, $g(\cdot,x)$ is a probability distribution over $\Omega$ and for any $B\in\mathcal{B}$, $P(X_2\in B|X_1)=g(B,X_1)$ a.s.. Then is it true that, for any measurable function $f$, \begin{align} \mathbb{E}[f(X_2)|X_1]=\mathbb{E}_{X_1}[f(X')], \end{align} where $\mathbb{E}_{x}$ denotes the expectation when $X'\sim g(\cdot,x)$?
I think that the statement is true when $X_1$ and $X_2$ take values in a countable space. If it is not true in general, is $\mathbb{E}_{x}f(X')$ at-least a measurable function from $\mathbb{R}\rightarrow\mathbb{R}$?