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# uncountable sums of positive numbers

The notion of sum of a series can be generalized to sums of nonnegative real numbers over arbitrary index sets.

let $I$ be a set and let $c$ be a mapping from $I$ to the nonnegative real numbers. Then we may define the sum as follows:

$\sum_{{i\in I}}c_{i}=\sup_{{s\subset I\atop\#s<\infty}}\sum_{{i\in s}}c_{i}$ |

In words, we are taking the supremum over all sums over finite subsets of the index set. This agrees with the usual notion of sum when our set is countably infinite, but generalizes this notion to uncountable index sets.

An important fact about this generalization is that the sum can only be finite if the number of elements $i\in I$ such that $c_{i}>0$ is countable. To demonstrate this fact, define the sets $s_{n}$ (where n is a nonnegative integer) as follows:

$s_{0}=\{i\in I\mid c_{i}\geq 1\}$ |

when $n>0$,

$s_{n}=\{i\in I\mid 1/n>c_{i}\geq 1/(n+1)\}$ |

If any of these sets is infinite, then the sum will diverge so, for the sum to be finite, all these sets must be finite. However, if these sets are all finite, then their union is countable. In other words, the number of indices for which $c_{i}>0$ will be countable.

This notion finds use in places such as non-separable Hilbert spaces. For instance, given a vector in such a space and a complete orthonormal set, one can express the norm of the vector as the sum of the squares of its components using this definition even when the orthonormal set is uncountably infinite.

This discussion can also be phrased in terms of Lesbegue integration with respect to counting measure. For this point of view, please see the entry support of integrable function with respect to counting measure is countable.

## Mathematics Subject Classification

40-00*no label found*

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## Comments

## generalisation

Can't this be extended to series of real numbers (maybe negative) as long as the series of |a_i| is convergent (in this sense)?

I think it can but maybe I am wrong.

My guess is that as long as the series is "absolutely convergent" (here in an extended sense), the sup of the series \sum a_i will also exist.

## Re: generalisation

"I think it can but maybe I am wrong."

Well, I am wrong!

the sup \sum a_i will exists but will not be the value of the sum

But it can still be defined.

sum only the positive elements (their sum exists because \sum |a_i| exists

sum only the |negatives| (also exists for the same reason)

subtract the second from the first. Thats the value of \sum a_i.

## Re: generalisation

You could do it for a general Banach space. If (a_i)_{i\in I} is a family of elements from a Banach space X, then for every finite subset F\subset I, define a_F=\sum_{i\in F}a_i. The set of all finite subsets (call it S) of I forms a directed set under inclusion. and hence (a_F)_{F\in S} forms a net in X. So we can say

\sum_{i\in I}a_i=\lim_{F\in S}a_F

if the limit of this net exists.

Something to note is that for a countable set I, this coincides with the usual notion of a summability only if the series converges absolutely.

## Re: generalisation

Here is an equivalent definiton (to bob1s) that uses less terminology.

Sum(c_i,i,I) = S

means

For all eps>0, there exists a finite subset t of I such that for all finite sets T such that t is a subset of T is a subset of I,

|S-(Sum(c_i,i,T))| < eps.