Example

— Let $X$ be a locally compact space, $t \mapsto \lambda_t$ a mapping of $T$ into the space $\mathscr{M}(X)$ of measures on $X$. To say that the family $t \mapsto \lambda_t$ is $\mu$-adequate means that it consists of positive measures and that the mapping $t \mapsto \lambda_t$ is scalarly essentially $\mu$-integrable and $\mu$-measurable for the topology $\sigma(\mathscr{M}(X), \mathscr{K}(X))$.\footnote{This is the definition of 'µ-adequate' in the first edition of Chapter 5 (Ch. V, 1st edn., §3, No. 1, Def. 1). In the second edition, of which the preceding chapter is a translation, the term 'µ-adequate' defines a weaker (more general) concept (Ch. V, §3, No. 1, Def. 1); more precisely, a scalarly essentially $\mu$-integrable family $t \mapsto \lambda_t$ of positive measures is $\mu$-adequate in the sense of the first edition of Chapter V if and only if it is $\mu$-adequate in the sense of the second edition and is vaguely $\mu$-measurable, that is, $\mu$-measurable for $\sigma(\mathscr{M}(X), \mathscr{K}(X))$ (Ch. V, §3, No. 1, Prop 2 b)). The two definitions are equivalent whenever the topology of $X$ has a countable base (ibid. Prop. 2 c)), which is the case for the applications in Chapter VI (§3); consequently for the rest of the chapter, no distinction is made between the two interpretations of 'µ-adequate', and the references to the first edition of Ch. V have been replaced by their nearest equivalent in the second edition.}

integral with respect to $\mu$ is the measure that was denoted $\int \lambda_t\, d\mu(t)$ in Ch. V, §3, No. 1.