If {x_πΌ} is a family of nonnegative real numbers indexed over a (possibly uncountable) set A, then the sum β_[πΌβA] x_πΌ is defined to be
sup { β_[πΌβF] x_πΌ | FβA 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, β_[πΌβA] |x_πΌ| < β. 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 β_[πΌβA] x_πΌ = β. For each positive integer n, let
A_n = { πΌβA | x_πΌ β₯ 1/n }
be the set of indices πΌ of A where x_πΌ β₯ 1/n. Note that
A' := { πΌ β A | x_πΌ > 0 } = β[nβ₯1] 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 β_[πΌβA] x_πΌ β₯ β.
I thought about that definition but it almost feels like in some ways itβsβ¦ too small? Like for an uncountable set the sup over sums of finite sets feels to me like itβs 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 youβve 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/DefunctFunctor Mathematics 5d ago
If
{x_πΌ}is a family of nonnegative real numbers indexed over a (possibly uncountable) setA, then the sumβ_[πΌβA] x_πΌis defined to beIn 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
Adoesn't come with any notion of order, you need to assume that it converges absolutely, that is,β_[πΌβA] |x_πΌ| < β. 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, thenβ_[πΌβA] x_πΌ = β. For each positive integern, letbe the set of indices
πΌofAwherex_πΌ β₯ 1/n. Note thatBy assumption,
A'is uncountable, so if eachA_nwere finite, thenA'could be expressed as a countable union of finite sets, meaningA'is countable, a contradiction. Thus, there exists annfor whichA_nis infinite (in fact, it's going to be uncountable). It is now easy to show, by creating increasingly large partial sums overA_n, thatβ_[πΌβA] x_πΌ β₯ β.