I would like to know under what conditions on $\rho$ - borel probability measure on $ X \times Y$ - the following statement holds true

$$ \int_{X \times Y} \varphi(x,y) \, d\rho = \int_X \left( \int_Y \varphi(x,y) \, d\rho(y|x) \right )d\rho_X \, ,$$

where $X$ is a compact domain or manifold in Euclidian space, $Y = \mathbb{R} $, $ \varphi : X \times Y \rightarrow \mathbb{R}$ is an integrable function, $\rho(y|x)$ denotes conditional probability measure w.r.t $x$ on $Y$ and $\rho_X$ marginal measure on $X$? While this equation is intuitively understandable for me I do not know what assumptions are necessary to made.

I also have trouble how to formally define $\rho(y|x)$ in the setting above.

  • $\begingroup$ Would referring to the the Radon-Nikodym derivative in an answer (en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem) be acceptable to you? $\endgroup$ – muaddib Jun 25 '15 at 0:18
  • $\begingroup$ I would really like to say "yes", but unfortunately I can't. I've read an wikipedia article but I fail to see how to apply it here. Is $\varphi$ ought to be Randon-Nikodem derivative? What to do with it next? Does it lead to Fubini Theorem somehow? How? Maybe my question should be restated: when $\rho$ could be broken into conditional probability measure and marginal measure and the relationship above holds? $\endgroup$ – ltw Jun 25 '15 at 13:50
  • $\begingroup$ Aha, I didn't mean is stating this is related to the Radon-Nikodym be sufficient to understand. Instead, would a full posted answer that uses the Radon-Nikodym derivative be readable. I need to assess what your knowledge base is. $\endgroup$ – muaddib Jun 25 '15 at 13:52
  • $\begingroup$ Well, it's hard to tell in advance. I'm a 5th year applied mathematics student. I did only standard courses on measure theory, probability theory, stochastic analysis etc. Less general answer would be helpful too - like an example of class of 'well-behaved' probability measures with properties that are more than enough for the statement from my post to hold. In the article where I found this statement authors told only that "a version of Fubini Theorem states that (...)" with no explanation which version. $\endgroup$ – ltw Jun 25 '15 at 14:30

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