Definition 1

— *Let $X$ be a locally compact space, $\Lambda : t \mapsto \lambda_t$ a scalarly essentially $\mu$-integrable mapping of $T$ into $\mathscr{M}_+(X)$, and $\nu$ the integral of $\Lambda$.

We say that $\Lambda$ is $\mu$-pre-adequate if, for every lower semi-continuous function $f \geq 0$ defined on $X$, the function $t \mapsto \int^\bullet f d\lambda_t$ is $\mu$-measurable on $T$ and*

We say that $\Lambda$ is $\mu$-adequate () if $\Lambda$ is $\mu'$-pre-adequate for every positive measure $\mu' \leq \mu$.*

It can be shown that if $\Lambda$ is $\mu$-pre-adequate and if the measure $\nu = \int \lambda_t d\mu(t)$ is moderated—in particular if $X$ is countable at infinity—then $\Lambda$ is $\mu$-adequate (Exer. 7); however, it is not known if these concepts are in general equivalent. In the statements in Nos. 2 and 3 below, the assertions preceded by an a) or a b) extend at once to pre-adequate mappings, whereas those preceded by a c) are valid only for adequate mappings.

(*) In the first edition, ' $\mu$-adequate' mappings were defined to be the scalarly essentially $\mu$-integrable and vaguely $\mu$-measurable mappings. The definition given here is more general (Prop. 2 below).

The following proposition often permits one to verify that a given mapping is $\mu$-adequate.