Corollary

— Suppose that $\mu$ is the sum of a summable family $(\mu_\alpha)_{\alpha \in A}$ of positive measures on $T$; in order that $\Lambda : t \mapsto \lambda_t$ be scalarly essentially $\mu$-integrable, it is necessary and sufficient that $\Lambda$ be scalarly essentially $\mu_\alpha$-integrable for every $\alpha \in A$ and that the family of measures $\int \lambda_t d\mu_\alpha(t)$ be summable. One then has

It follows immediately that every scalarly essentially $\mu$-integrable mapping is also scalarly essentially $\mu'$-integrable for every measure $\mu' \leq \mu$.

In this section we shall limit ourselves to the study of scalarly essentially integrable mappings of $T$ into $\mathscr{M}_+(X)$ that have the property contemplated in the following definition: