Exercise 6, § 3

a) Let $I = [0, 1]$. Let $f$ be a continuous function $\geq 0$ on $\mathbf{R}$, with support contained in $[-1, 0]$. Show that the set of functions $\gamma(s)f|I\ (s \in I)$ has infinite rank in $\mathscr{K}(I)$.

b) Let $f_1, \ldots, f_n$ be in $\mathscr{K}(\mathbf{R})$. Let $M$ be the set of measures $\mu \in \mathscr{M}(I)$ such that $\mu(f_1) = \ldots = \mu(f_n) = 0$. Show that the set of functions $y \mapsto \int f(x + y)\ d\mu(x)\ (y \in I)$, where $\mu$ runs over $M$, has infinite rank in $\mathscr{K}(I)$. (Make use of a.)

c) Let $g_1, \ldots, g_p$ be in $\mathscr{K}(\mathbf{R})$. Deduce from b) that there exist $\mu \in M$ and $\nu \in \mathscr{M}(I)$ such that $\nu(g_1) = \ldots = \nu(g_p) = 0$, $(\nu * \mu)(f) \neq 0$.

d) Deduce from c) that the mapping $(\mu, \nu) \mapsto \nu * \mu$ of $\mathscr{M}(T) \times \mathscr{M}(T)$ into $\mathscr{M}(T)$ is not vaguely continuous.