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I have studied fundamental property of conditional expectation:

$\mathbb{E}[r(X)\mathbb{E}(Y|X)] = \mathbb{E}(r(X)Y] \qquad \text{for every function} \qquad r:S \rightarrow R$

My question is what does mean function $r$. Why left side of equation equals to the right side and what is intution behind this function?

Thanks in advance

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    What is the meaning of the sets $S$ and $R$ ? – 2017-02-19
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    You may refer to the lower half of [my answer to another probability problem](http://math.stackexchange.com/a/2148427/290189) for a similar (though not exact) proof. – 2017-02-19
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    Note that we can think of $(X,Y)$ as a random variables that takes values in a subset of $S \times T$ http://www.math.uah.edu/stat/expect/Conditional.html – 2017-02-19

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I'm a little confused about the domain/range $S\to R...$ I would think it would just be from the reals to the reals. But, anyway, by function, they mean... function. I don't know another way to put it. Just a function like $r(X)=X^2$ or any other example. The key is that the property holds for any function.

The quickest way to see that the left side equals the right is that $$ r(X)E(Y|X) = E(r(X)Y|X)$$ since you can always pull something conditioned on out of a conditional expectation (when it's conditioned on, it's effectively constant). So plugging this in, we have $$E(r(X)E(Y|X))=E(E(r(X)Y|X)) = E(r(X)Y) $$

where the second equality is the law of total expectation.

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    Actually, $S$ may be any measurable space (the one $X$ takes its values in) and $R$ must be $\mathbb R$ the real line. – 2017-03-08