Exercise 18, § 4

Notations and conventions are those of Exer. 15.

a) Let $\mu$ be a bounded measure on G, whose support contains at least two distinct points. Show that there exists a compact set K, and a number k such that $0 < k < 1$, for which $\| \varphi_{sK} \cdot \mu \| \leq k \| \mu \|$ for all $s \in G$. (If $t, t'$ are two distinct points in the support of $\mu$, take for K a neighborhood of e sufficiently small that sK cannot intersect both tK and t'K.)

b) Let $\mu$ be a bounded measure $\geq 0$ on G, whose support contains at least two distinct points. Show that there exists a function $f \in L^\infty(G, \beta)$, not equivalent to a function $\geq 0$, such that $\mu * f$ is $\geq 0$ locally almost everywhere for $\beta$. (Use a), taking f to be equal to $-1$ on K and to a suitable positive constant on $\mathbf{C}K$.) If, in addition, the support of $\mu$ is compact, there exists a function $f$ having the preceding properties and having compact support.

c) Let $\mu_1, \mu_2$ be two nonzero, bounded positive measures on G, permutable for convolution, such that the support K of $\mu_1$ is compact and contains e. Set $\mu = \mu_1 + \mu_2$. Let V be a symmetric compact neighborhood of e containing K, g the function equal to $\mu * \varphi_V$ on $\mathbf{C}V$, and to 0 on V; show that the function $\mu * (g - (\mu_1 * \varphi_V))$ is $\geq 0$ locally almost everywhere in $\mathbf{C}(V^2)$. On the other hand, show that there exists a compact set H such that the function $\mu_2 * \varphi_H$ is $\geq 0$ almost everywhere in $V^2$.

d) Deduce from c) that if $\mu$ is a bounded positive measure on $G$ whose support contains at least two distinct points, there exists a function $f$ that belongs to every $L^p(G, \beta)$ ($1 \leq p \leq +\infty$), is not equivalent to a function $\geq 0$, and is such that $\mu * f$ is $\geq 0$ locally almost everywhere, when one makes in addition one of the following hypotheses: $\alpha)$ $G$ is abelian; $\beta)$ there is a point $a \in G$ such that $\mu(\{a\}) > 0$. (Suitably decompose $\mu$ as a sum of two permutable measures $\geq 0$.)