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?