Exercise 2, § 1

Let T, X, Y be three locally compact spaces, with Y having a countable base. Let $\mu$ be a measure $\geq 0$ on T, $t \mapsto \lambda_t$ ($t \in T$) a $\mu$-adequate$^1$ family of measures $\geq 0$ on X, and set $\nu = \int \lambda_t d\mu(t)$. Let $x \mapsto \rho_x$ ($x \in X$) be a $\nu$-adequate family of measures $\geq 0$ on Y. Suppose, moreover, that one of the following two conditions is satisfied: a) X is countable at infinity; b) $\nu$ is bounded. Under these conditions, show that there exists a locally $\mu$-negligible set $N'$ such that, for every $t \notin N'$, the family $x \mapsto \rho_x$ is $\lambda_t$-adequate; moreover, the family $t \mapsto \int \rho_x d\lambda_t(x)$, defined for $t \notin N'$, is $\mu$-adequate, and

(To prove that $x \mapsto \rho_x$ is scalarly $\lambda_t$-integrable locally almost everywhere, make use of Lemma 1 of §3, No. 1, using, in case b), the fact that $\lambda_t$ is bounded locally almost

(1) In the sense of the first edition of Ch. V (cf. the footnote to §1, No. 1, Example); similarly for Exercises 3 and 4.

everywhere; same method for seeing that $t \mapsto \int \rho_x d\lambda_t(x)$ is scalarly $\mu$-integrable; use also Prop. 13 of No. 5.)