Exercise 5, § 1

Let $(\mu_n)$ be a sequence of positive measures on T that converges quasi-strongly to $\mu$ in $\mathscr{M}(T)$ (Ch. III, §1, Exer. 8).

a) Show that if a mapping $f$ of T into a topological space G is $\mu_n$-measurable for every $n$, then it is $\mu$-measurable (make use of Exercise 6 b) of Ch. IV, §1).

b) Let $f$ be a mapping of T into a Banach space F, that is essentially $\mu_n$-integrable for every $n$. Show that if the sequence $(\mu_n^* (|f|))$ is bounded, then $f$ is essentially $\mu$-integrable (Exer. 2). Show by means of an example that one does not necessarily have $\int f \, d\mu = \lim_{n \to \infty} \int f \, d\mu_n$ (take for $\mu_n$ a point measure on a compact space, such that $\lim \| \mu_n \| = 0$); however, this relation holds when $f$ is bounded and has compact support.