Exercise 11, § 3

Let $t \mapsto \lambda_t$ be a $\mu$-adequate mapping of T into $\mathscr{M}_+(X)$ satisfying the following condition: for every compact subset K of T, there exists a compact subset $L_K$ of X such that $\operatorname{Supp}(\lambda_t) \subset L_K$ for all $t \in K$.

a) Show that for every function $f \geq 0$ defined on X,

(make use of Prop. 3 a)).

b) If $f \geq 0$ is locally $\nu$-negligible, then the set of $t \in T$ such that $f$ is not locally $\lambda_t$-negligible is locally $\mu$-negligible.

c) If $f$ is a $\nu$-measurable mapping of X into a topological space X, then the set of $t \in T$ such that $f$ is not $\lambda_t$-measurable is locally $\mu$-negligible.

d) For every $\nu$-measurable function $f \geq 0$, the mapping $t \mapsto \int^* f\, d\lambda_t$ is $\mu$-measurable and

e) Let $\mathbf{f}$ be a function defined on X, with values in a Banach space or in $\overline{\mathbf{R}}$, and essentially $\nu$-integrable. Show that the set of $t \in T$ such that $\mathbf{f}$ is not $\lambda_t$-integrable is locally $\mu$-negligible, the function $t \mapsto \int \mathbf{f}(x)\, d\lambda_t(x)$ (defined locally almost everywhere for $\mu$) is essentially $\mu$-integrable, and