Exercise 8, § 3

Let $\Lambda : t \mapsto \lambda_t$ be a mapping of T into $\mathscr{M}_+(X)$. a) Let $\mu_1, \ldots, \mu_n$ be measures on T, and $\mu = \mu_1 + \cdots + \mu_n$. For $\Lambda$ to be $\mu$-adequate, it is necessary and sufficient that $\Lambda$ be $\mu_i$-adequate for every $i$ (make use of the decomposition lemma). b) Suppose that $\mu$ is the supremum of an increasing directed family $(\mu_i)_{i \in I}$. For $\Lambda$ to be $\mu$-adequate, it is necessary and sufficient that $\Lambda$ be $\mu_i$-adequate for all $i \in I$ and scalarly essentially $\mu$-integrable. c) Suppose that $\mu$ is the sum of a summable family $(\mu_j)_{j \in J}$ of positive measures. For $\Lambda$ to be $\mu$-adequate, it is necessary and sufficient that $\Lambda$ be $\mu_j$-adequate for all $j \in J$ and that $\Lambda$ be scalarly essentially $\mu$-integrable. d) For $\Lambda$ to be $\mu$-adequate, it is necessary and sufficient that $\Lambda$ be scalarly essentially $\mu$-integrable and that $\Lambda$ be $\mu'$-pre-adequate for every measure $\mu' \leq \mu$ with compact support.