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Are $\displaystyle \sum_{\large x\in S}f(x)$ and $\displaystyle \prod_{\large x\in S}f(x)$ valid?
Are there any standard notations for for the sum/product of a function over the elements of a set.

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  • $\begingroup$ It depends on the fact that if $S$ is countable or not. Your notation is standard when $S$ is uncountable. $\endgroup$ – Albert Dec 23 '15 at 17:04
  • $\begingroup$ This is the standard notation, but it only makes sense if S is a finite set or when the function takes only finitely many non zero values. $\endgroup$ – Arun Kumar Dec 23 '15 at 17:07
  • $\begingroup$ If $f \ge 0$ then one can define $\sum_{x \in S} f(x) = \sup_{F \subset S, F \text{ finite}} \sum_{x \in F} f(x)$. This works for uncountable as well. $\endgroup$ – copper.hat Dec 23 '15 at 18:28
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These are perfectly valid notations and can/are being used typically for uncountable sets or for finite countable set. However, for infinite uncountable sets, in some cases care needs to be taken. For instance, no unique value can be assigned to the sum $$\sum_{n \in \mathbb{N}} \dfrac{(-1)^n}n$$ For an infinite countable set $S$, it is essential to index the set in a certain order to associate a unique value to the sum.

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