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I was fascinated when I heard that the most intuitive laws of arithmetic are no longer necessarily valid when it comes to the sum of an infinite sequence, which be denoted by $$S = \sum_{n=0,1,2,...}a_n$$ There are different ways of convergence, and particularly interesting I find Riemann's rearrangement theorem, which states that for conditionally convergent series, it is possible to rearrange the series to converge to any given value.

Since this is all about the sum of a countably infinite set of numbers, I wonder whether there is a similar field of study on the sums of uncountably infinite sets of numbers, which could be written like this:

$$S = \sum_{x\in\mathbb{R}}a_x$$ Or if interpreted as the sum of all values of a function $f(x)$: $$S = \sum_{x\in\mathbb{R}} f(x)$$

Some minor observations are that:

  • if $f$ is continuous and $f(x)\ge 0$ for $x\in [a,b]$ then $S_{[a,b]}=\sum_{x\in [a,b]}f(x) \to +\infty$ (and the opposite if $f(x)\le0$ in $[a,b]$)

  • The integral of $f(x)$ in $[a,b]$ seems to be the sum $S_{[a,b]}$ diveded by the number of already counted values of $f$, however this seems not quite right because we are talking about an uncountably infinite amount of numbers and we have no notion of transitioning from countably to uncountably infinite values.

Is there any study on the sum of uncountably infinite sets of numbers?

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marked as duplicate by Jack, Ittay Weiss, Adam Hughes, C. Falcon, Brian M. Scott sequences-and-series Dec 8 '16 at 19:33

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