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Let $S$ be an uncountable set of indexed real numbers. So the same number can occur more than once in the set (although with a different index). I don't assume that there is any ordering on $S$. Is there a simple necessary and sufficient condition for $S$ to have a well-defined average?

I take it that a sufficient condition for there to be a well-defined average is if all but a finitely many of the indexed numbers are identical. Is this necessary as well?

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    What's the average value of all real numbers (an uncountable set)? There are some power law probability distributions, eg Cauchy, that converge so slowly that even first moment (mean) doesn't converge.2012-12-18
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    Quite interesting question. All but countably many should be enough. The challenge would be to pin down the question enough to be able to give a precise answer. For sets, as opposed to multisets, there is a literature.2012-12-18

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