Proposition 7

For the family $(\mu_i)_{i \in I}$ to be summable, with sum $\mu$, it is necessary and sufficient that the encumbrance $p = \sum_{i \in I} \mu_i^\bullet$ be locally bounded, in which case $p = \mu^\bullet$. For every compact subset K of T, the family $((\mu_i)_K)_{i \in I}$ is then summable in $\mathscr{M}(K)$, and $\mu_K = \sum_{i \in I} (\mu_i)_K$.

With notations as in Def. 9, $\lambda_\alpha^\bullet = \sum_{i \in \alpha} \mu_i^\bullet$ for every finite subset $\alpha$ of I (No. 2, Remark 1). The statement is then an immediate consequence of Prop. 6.

The relation $\mu_K = \sum_{i \in I} (\mu_i)_K$ and Prop. 2 of Ch. V, §2, No. 2 yields the following result: