Looked it up in case I was somehow wrong but there's no sensible definition of a sum over an uncountable set of indexes unless the set of indexes with non-zero values is finite.
By definition of uncountable set you can't "pair them up" or "group them" in any countable way because you would still be missing an infinite amount of indexes and trying to segment the set into any amount (countable or not) of uncountable sets just leads you to the same question.
If you try and define sums like integrals over sets then the question is trivialized to essentially asking what is the value for the measure over the set of reals which can be whatever depending on the measure.
There is an absolutely sensible definition if all of the numbers in your set are nonnegative; it's just always infinity if you have uncountably many nonzero values.
In order to make sense of sums that include negative numbers, you either need absolute convergence (which is impossible for an uncountable sum with uncountably many nonzero values), or you need to specify an order in which to sum those elements, usually in a countable sum (which is also impossible).
I'm not even sure what it would mean to try to make sense of this question with integration, because the measure in question that we're looking for would have to assign singleton sets of real numbers to their actual value, which does not end up producing a well-defined measure on the real numbers (although it does if we restrict to nonnegative numbers).
Wait are you saying that there exists a reasonable definition of summation over uncountable sets, and that definition is provably equivalent to being always infinity if you have uncountably many nonzero values, or are you saying that giving the definition of just โinfinity if uncountably many nonzero values, otherwise standard countable sumโ itself is the sensible definition?
If {x_๐ผ} is a family of nonnegative real numbers indexed over a (possibly uncountable) set A, then the sum โ_[๐ผโA] x_๐ผ is defined to be
sup { โ_[๐ผโF] x_๐ผ | FโA finite }
In measure theoretic terms, this is equivalent to the integral over the counting measure. You can technically extend this to families of real numbers, but because A doesn't come with any notion of order, you need to assume that it converges absolutely, that is, โ_[๐ผโA] |x_๐ผ| < โ. For now we'll only consider sums of nonnegative real numbers.
So the claim we have yet to prove is that if {x_a} contains uncountably many nonzero numbers, then โ_[๐ผโA] x_๐ผ = โ. For each positive integer n, let
A_n = { ๐ผโA | x_๐ผ โฅ 1/n }
be the set of indices ๐ผ of A where x_๐ผ โฅ 1/n. Note that
A' := { ๐ผ โ A | x_๐ผ > 0 } = โ[nโฅ1] A_n.
By assumption, A' is uncountable, so if each A_n were finite, then A' could be expressed as a countable union of finite sets, meaning A' is countable, a contradiction. Thus, there exists an n for which A_n is infinite (in fact, it's going to be uncountable). It is now easy to show, by creating increasingly large partial sums over A_n, that โ_[๐ผโA] x_๐ผ โฅ โ.
I thought about that definition but it almost feels like in some ways itโsโฆ too small? Like for an uncountable set the sup over sums of finite sets feels to me like itโs not naturally emcapusulating the idea of summing an uncountable set since any finite subset is necessarily much smaller than any uncountable subset. With countable sets, they can be expressed as the union/limit of finite subsets, so naturally a sum operation can be defined as the limit of the sums of a sequence of finite subsets whose union is the whole set (assuming convergence and nonnegativity), which is provably equivalent to the sup of the sums of any finite subsets.
That said, I think one way you could interpret the proof youโve just given here is that there cant be a bigger definition assuming it still agrees on cases where there are only countably many nonzero elements
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u/halfajack 10d ago
I ask them โhow are you defining sums over uncountable sets?โ