uncountable sums of positive numbers

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

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

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

An important fact about this generalization is that the sum can onlybe finite if the number of elements $i\in I$ such that $c_i>0$ iscountable. To demonstrate this fact, define the sets $s_n$ (where nis a nonnegative integer) as follows:

$$s_0=\{i\in I\mid c_i\ge 1\}$$

when $n>0$,

$$s_n=\{i\in I\mid 1/n>c_i\ge 1/(n+1)\}$$

If any of these sets is infinite, then the sum will diverge so, forthe sum to be finite, all these sets must be finite. However, ifthese sets are all finite, then their union is countable. In otherwords, 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 completeorthonormal set, one can express the norm of the vector as the sum ofthe squares of its components using this definition even when theorthonormal set is uncountably infinite.

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