Exercise 5, § 4

Let G be a locally compact group countable at infinity, $\beta$ a left Haar measure on G, $a \in \mathbf{R}_+^*$, and $(\nu_u)_{0 \leq u \leq a}$ a family of positive measures on G satisfying the following conditions: (i) $\nu_0 = \varepsilon_e$; $\nu_{u+v} = \nu_u * \nu_v$ if $u + v \leq a$; $\nu_u = \nu_u^\vee$; (ii) for $0 < u \leq a$, $\nu_u = f_u \cdot \beta$ with $f_u \geq 0$ lower semi-continuous; (iii) $\nu_u$ is a vaguely continuous function of $u$. a) Let $f \in \mathscr{K}_+(G)$. Show that, for $0 < u \leq a$,

b) Let $f \in \mathscr{K}(G)$. Show that $f_{u/2} * f$ has $\beta$-integrable square for $0 < u \leq a$, and that (1) again holds.

c) Let $f \in \mathscr{K}(G)$ be such that $\nu_a * f = 0$. Show that $f = 0$. (One has $\nu_{a/2^n} * f = 0$ for every integer $n > 0$ by b), therefore $f = 0$ by §3, No. 3, Remark 1).)

d) Let $\nu \in \mathscr{C}'(G)$ be such that $\nu_a * \nu = 0$. Show that $\nu = 0$. (Regularize $\nu$ by functions in $\mathscr{K}(G)$ and apply c).)