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I stumbled upon an interesting looking work by R. Yeung that shows an interesting relation between information and measure theory called A New Outlook on Shannon’s Information Measures. In this work the author frequently uses the notion of a "set variable $\tilde{X}$ corresponding to a random variable $X$". I have never heard of such a notion.

Unfortunately, for the author it seems to be common knowledge as he does not explain it. Even more frustrating, looking for such a notion only lead me to other articles by the same author and in none of them I found an explanation.

Does someone know what is meant by "a set variable corresponding to a random variable" or, even better, can point me to a reference where such a notion is explained?

Thank you very much in advance!

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  • $\begingroup$ Given that the author takes the union of set variables, I suspect they mean by a "set variable" some confusing superposition of the name for the set of values, and a symbol that is regarded as ranging somehow across that set of values. $\endgroup$
    – LSpice
    Feb 23, 2022 at 15:11
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    $\begingroup$ Thank you @LSpice for your reply! From what the author claims in his work, e.g. $I(X;Y) = \mu(X \cap Y)$, I can deduce that he means something deeper than $\tilde{X}=X(\Omega)$ , as there are clearly $X$ and $Y$ with positive mutual information but $X(\Omega) \cap Y(\Omega) = \varnothing$. $\endgroup$
    – joemrt
    Feb 23, 2022 at 15:21
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    $\begingroup$ Turns out there is a wikipedia article on this. Although I can't say the article explains it in sufficient detail for me, I have now the impression as if the "set variable" is nothing but an abstract representation - you can pick whatever you like as long as you satisfy some basic conditions. $\endgroup$
    – joemrt
    Feb 23, 2022 at 16:00

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