Proposition 1

Suppose that μ is the supremum of an increasing directed family $(\mu_i)_{i \in I}$ of positive measures on T; in order that $\Lambda : t \mapsto \lambda_t$ be scalarly essentially μ-integrable, it is necessary and sufficient that Λ be scalarly essentially $\mu_i$-integrable for all $i \in I$ and that the family $(\int \lambda_t\, d\mu_i(t))_{i \in I}$ be bounded above in $\mathscr{M}(X)$. One then has,

For, verifying that $\Lambda$ is scalarly essentially integrable for a positive measure $\eta$ on $T$ comes down to verifying that $t \mapsto \lambda_t(g)$ is $\eta$-measurable and admits a finite essential upper integral, with respect to $\eta$, for every function $g \in \mathscr{H}_+(X)$. The proposition therefore follows at once from Prop. 11 of §1, No. 4 and its Corollary 2.