Remark 1

When $I$ is finite, Definition 1 reduces to the ordinary definition of the sum of a finite family. More generally, if $I$ is arbitrary and $x_i = 0$ except when the index $i$ belongs to a finite subset $J$ of $I$, then the sum $\sum_{i \in I} x_i$ is equal to $\sum_{i \in J} x_i$.