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Suppose that $y$ is a random variable and $X_1$ and $X_2$ random vectors. Let $Z$ be the random vector that collects the distinct elements in $X_1$ and $X_2$. Can you please explain the difference (if any) between $$ E(y|X_1,X_2)\quad\text{and}\quad E(y|Z)? $$ If you can, also please point me to an accessible reference so that I am better equipped to overcome this sort of confusion. I am familiar with the basics of measure theory and the measure-theoric definition of conditional expectations. Thank you.

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    $\begingroup$ None. $ $ $ $ $ $ $\endgroup$ – Did Nov 17 '14 at 14:55
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    $\begingroup$ @Did Can you please elaborate, even by writing a short proof? I have always found the measure theoric definition hard to apply to practical situation like this. $\endgroup$ – yurnero Nov 17 '14 at 14:56
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    $\begingroup$ Sure: the sigma-algebras $\sigma(X_1,X_2)$ and $\sigma(Z)$ coincide. $\endgroup$ – Did Nov 17 '14 at 14:58

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