Exercise 19, § 3

The notations and conventions are those of Exercise 15.

a) Now prove that if a bounded positive 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 commutative (use Exercise 18) (*). For $p = +\infty$, the same conclusion holds without any additional assumption on $G$.

b) Let $\mu$ be a bounded positive measure on $G$ such that: $1^o$ $\gamma(\mu)$ is a surjective endomorphism of $L^1(G, \beta)$ ; $2^o$ for every function $f \in L^1(G, \beta)$ the relation $\mu * f \geq 0$ locally almost everywhere implies

(*) For an example where $\mu$ is non-point and such that $\mu * f \geq 0$ implies $f \geq 0$, see Williamson, Proc. Edimb. Math. Soc., 1957, pp. 71-77.

$f \geqslant 0$ locally almost everywhere. Now prove that $\mu$ is a point measure. (Observe that for every function $g \in L^\infty(G, \beta)$, the relation $\tilde{\mu} * g \geqslant 0$ locally almost everywhere implies $g \geqslant 0$ locally almost everywhere).