Definition 9

Let $(\mu_i)_{i \in I}$ be a family of measures on T. Let A be the set of finite subsets of I; for every $\alpha \in A$ let $\lambda_\alpha = \sum_{i \in \alpha} \mu_i$. If the family $(\lambda_\alpha)$ admits a supremum $\mu$ in $\mathscr{M}(T)$, the family $(\mu_i)$ is said to be summable, $\mu$ is called the sum of the family $(\mu_i)$, and one writes $\mu = \sum_{i \in I} \mu_i$.

This definition extends the definition of Ch. V, §2, No. 1.