Remark

The theorem of associativity (Chapter III, § 5, no. 3, Theorem 2) remains valid when G is replaced by $\overline{\mathbf{R}}$ and the $x_i$ are assumed to be $\geq 0$. This is clear if $\sum_{i \in I} x_i$ is finite. Suppose, on the contrary, that $\sum_{i \in I} x_i = + \infty$. Then for each finite $a > 0$ there is a finite subset H of I such that $\sum_{i \in H} x_i \geq a$. Let K be a finite subset of L such that $H \subset \bigcup_{\lambda \in K} I_\lambda$; then since $s_\lambda \geq \sum_{i \in I_\lambda \cap H} x_i$ for all $\lambda \in K$, we have

which shows that $\sum_{\lambda \in L} s_\lambda = + \infty$. We leave to the reader the task of formulating the analogous proposition for multipliable families of numbers in $[0, 1]$ or in $[1, + \infty]$.