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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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closed as unclear what you're asking by Did, Davide Giraudo, drhab, BLAZE, Willie Wong Nov 9 '15 at 21:35

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

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