Is it possible by some means to define a notion of sum over the elements of an uncountable set $S$ of real numbers.I thought of something like $\sum_{\alpha \in \tau } a_\alpha:=\sup_{T\subset S ,|T|=\aleph_0}\sum_{a_{\alpha}\in T}a_\alpha$ where $\tau$ is the index set.Does this makes much sense?
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If it is a set of non negative real numbers then we can define the sum as the supremum over the sums of the elements of all finite subsets. However, this has a chance to be a finite number only when the number of non zero elements in the set is at most countable. – Mark Dec 28 '19 at 14:51
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Or this: Does uncountable summation, with a finite sum, ever occur in mathematics? – Martin R Dec 28 '19 at 14:52
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To make sure the sum over $T$ is well-defined, you want to pick $T$ finite. In general we can define $\sum_{\alpha\in \tau} a_\alpha = c$ if for every $\varepsilon>0$ exists a finite set $T_\varepsilon \subset \tau$ such that for every finite set $T\subset S$ with $T_\varepsilon \subseteq T$ holds $$ \Vert c - \sum_{\alpha\in T} a_\alpha \Vert <\varepsilon.$$ – Severin Schraven Dec 28 '19 at 14:59