Exercise 10, § 5

a) Let $\mu$ be a diffuse positive measure $\neq 0$ on a locally compact space $T$. Show that there exists in $\mathscr{L}^1(T, \mu)$ a sequence of functions $f_n$ such that $N_1(f_n) = 1$ for all $n$ and such that the sequence of diffuse measures $f_n \cdot \mu$ converges vaguely to a point measure $\varepsilon_a$. From this, deduce that $L^1(T, \mu)$ (hence also $L^\infty(T, \mu)$) is not reflexive.

b) Let $\nu$ be an atomic positive measure on $T$, whose support is infinite. Let $A$ be the smallest set carrying $\nu$, $B$ a countable infinite subset of $A$, and $\lambda$ the measure $\varphi_B \cdot \nu$. Show that the dual of $L^\infty(T, \lambda)$ is not separable, hence cannot be isomorphic to $L^1(T, \lambda)$ (cf. TVS, I, §2, Exer. 1).

c) Deduce from a) and b) that, for a positive measure $\mu$ on a locally compact space $T$ to be such that $L^1(T, \mu)$ is reflexive, it is necessary and sufficient that the support of $\mu$ be finite (make use of Exer. 9).