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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 $\mathbb{R}\rightarrow\mathbb{R}$?

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If $g$ were a kernel, we would assume that $x\mapsto g(B,x)$ is measurable from $\mathbb{R}$ to itself for every fixed $B\in{\cal B}$. You have not assumed this, and I can't tell if this is a deliberate omission or an oversight. Could you clarify please? –  Byron Schmuland Jan 27 '13 at 23:23
OP: Why don't you answer @Byron's query? –  Did Jan 30 '13 at 11:46

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