Exercise 18, § 3

The notations and conventions are those of Ex. 15.

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

b) Let $\mu$ be a bounded measure $\geq 0$ on $G$, whose support contains at least two distinct points. Now prove that there exists a function $f \in L^\infty(G, \beta)$, not equivalent to a function $\geq 0$, and such that $\mu * f$ is $\geq 0$ locally almost everywhere for $\beta$. (Use a), taking $f$ equal to $-1$ in $K$ and to a suitable positive constant in $C_K$.)

If moreover 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 positive bounded measures on $G$, permutable for convolution, and such that the support $K$ of $\mu_1$ is compact and contains $e$. Put $\mu = \mu_1 + \mu_2$. Let $V$ be a compact symmetric neighbourhood of $e$ containing $K$, $g$ the function equal to $\mu * \varphi_V$ in $C_V$, to $0$ in $V$; now prove that the function $\mu * (g - (\mu_1 * \varphi_V))$ is locally almost everywhere $\geq 0$ in $C(V^2)$. On the other hand, now prove that there exists a compact set $H$ such that the function $\mu_2 * \varphi_H$ is almost everywhere $\geq 0$ in $V^2$.

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