Exercise 19, § 4

Notations and conventions are those of Exer. 15. a) Show that if a positive bounded measure $\mu$ on $G$ is such that for every function $f \in L^p(G, \beta)$ ($p$ given, $1 \leq p < \infty$) the relation $\mu * f \geq 0$ locally almost everywhere implies $f \geq 0$ locally almost everywhere, then one can conclude that $\mu$ is a point measure in each of the following cases: $\alpha)$ $G$ is compact; $\beta)$ $G$ is discrete; $\gamma)$ $G$ is abelian (use Exer. 18) (*). For $p = +\infty$, the same conclusion is valid without a supplementary hypothesis on $G$. b) Let $\mu$ be a positive bounded measure on $G$ such that: $1^\circ$ $\gamma(\mu)$ is a surjective endomorphism of $L^1(G, \beta)$; $2^\circ$ for every function $f \in L^1(G, \beta)$, the relation $\mu * f \geq 0$ locally almost everywhere implies $f \geq 0$ locally almost everywhere. Show that $\mu$ is then a point measure. (Observe that for every function $g \in L^\infty(G, \beta)$, the relation $\overset{\vee}{\mu} * g \geq 0$ locally almost everywhere implies $g \geq 0$ locally almost everywhere.)