Exercise 11, § 3

Let G be a locally compact group; on $\mathscr{M}^1(G)$, let us denote by $\mathscr{T}_I$ the topology $\sigma(\mathscr{M}^1(G), \mathscr{K}(G))$, by $\mathscr{T}_{III}$ the topology $\sigma(\mathscr{M}^1(G), C^\infty(G))$, and let us write $\mathscr{M}_I, \mathscr{M}_{III}$ for the space $\mathscr{M}^1(G)$ equipped respectively with $\mathscr{T}_I$ and $\mathscr{T}_{III}$.

a) Let A be a bounded subset of $\mathscr{M}_I$, and B a relatively compact subset of $\mathscr{M}_{III}$. Show that the restriction to $A \times B$ of the mapping $(\mu, \nu) \mapsto \mu * \nu$ of $\mathscr{M}_I \times \mathscr{M}_{III}$ into $\mathscr{M}_I$ is continuous (use Exer. 10 to reduce to evaluating an integral $\iint f(st)\ d\mu(s)\ d\nu(t)$ when $f(st) = \sum_i u_i(s)v_i(t)$ with $u_i$ and $v_i$ in $\mathscr{K}(G)$).

b) Give an example where G is compact (hence $\mathscr{T}_I = \mathscr{T}_{III}$) showing that $(\mu, \nu) \mapsto \mu * \nu$, regarded as a mapping of $\mathscr{M}_I \times \mathscr{M}_{III}$ into $\mathscr{M}_I$, is not hypocontinuous with respect to the relatively compact subsets of $\mathscr{M}_I$, nor with respect to the relatively compact subsets of $\mathscr{M}_{III}$ (cf. Exer. 6).

c) Let A, B be two relatively compact subsets of $\mathscr{M}_{III}$. Show that the restriction to $A \times B$ of the mapping $(\mu, \nu) \mapsto \mu * \nu$ of $\mathscr{M}_{III} \times \mathscr{M}_{III}$ into $\mathscr{M}_{III}$ is continuous (same method as in a)).

d) Denote by E the subspace of $\mathbf{R}^G$ formed by the linear combinations of characteristic functions of open subsets of G, by $\mathscr{T}_{IV}$ the topology $\sigma(\mathscr{M}^1(G), E)$, and by $\mathscr{M}_{IV}$ the space $\mathscr{M}^1(G)$ equipped with $\mathscr{T}_{IV}$. Recall that the topologies induced by $\mathscr{T}_{III}$ and $\mathscr{T}_{IV}$ on a bounded subset of $\mathscr{M}^1(G)$ consisting of positive measures, are in general distinct (Ch. V, §5, Exer. 16 c)). Recall also that the compact subsets of $\mathscr{M}_{IV}$ are the same as those of $\mathscr{M}^1(G)$ for the weakened topology $\sigma(\mathscr{M}^1(G), (\mathscr{M}^1(G))')$ on the Banach space $\mathscr{M}^1(G)$ (Ch. VI, §2, Exer. 12). Show that if A, B are two relatively compact subsets of $\mathscr{M}_{IV}$, the restriction to $A \times B$ of the mapping $(\mu, \nu) \mapsto \mu * \nu$ of $\mathscr{M}_{IV} \times \mathscr{M}_{IV}$ into $\mathscr{M}_{IV}$ is continuous. (Restrict attention to the case that A and B are compact, and begin by proving that the image of $A \times B$ under the preceding mapping is then compact for $\sigma(\mathscr{M}^1(G), (\mathscr{M}^1(G))')$; for this, apply the theorems of Eberlein and Šmulian (TVS, IV, §5, No. 3), and thus reduce to proving that if $(\mu_n), (\nu_n)$ are two sequences that converge to 0 in $\mathscr{M}_{IV}$, then the same is true of the sequence $(\mu_n * \nu_n)$; make use of Prop. 12 and Exer. 15 of Ch. V, §5. Finally, to prove that $(\mu, \nu) \mapsto \mu * \nu$ is continuous for $\mathscr{T}_{IV}$, use c) and the fact that $\mathscr{T}_{III}$ is coarser than $\mathscr{T}_{IV}$.)

e) Take $G = \mathbf{R}^2$; let $a, b$ be the vectors of the canonical basis of G over $\mathbf{R}$, $\rho_n$ the measure on the interval $I = [0, \pi]$ of $\mathbf{R}$ having as density with respect to Lebesgue measure the function $\sin(2^n x)$, and $\mu_n$ the measure $\rho_n \otimes \varepsilon_0$ on $\mathbf{R} \times \mathbf{R}$; on the other hand let $\nu_n = \varepsilon_{b/2^n} - \varepsilon_0$ on G. Show that the sequence $(\mu_n)$ tends to 0 in $\mathscr{M}_{IV}$ and that the sequence $(\nu_n)$ tends to 0 in $\mathscr{M}_{III}$, but that the sequence $(\mu_n * \nu_n)$ does not tend to 0 in $\mathscr{M}_{IV}$ (cf. Ch. V, §5, Exer. 16).

f) Give an example where G is compact and where $(\mu, \nu) \mapsto \mu * \nu$, regarded as a mapping of $\mathscr{M}_{IV} \times \mathscr{M}_{IV}$ into $\mathscr{M}_{IV}$, is not hypocontinuous relative to the compact subsets of $\mathscr{M}_{IV}$ (same method as in Exer. 6, on observing that for a given $f \in E$ there exist compact subsets $H \subset \mathscr{M}_{IV}$ such that the set of functions $t \mapsto \int f(st)\ d\mu(s)$, where $\mu$ runs over H, has finite rank over $\mathbf{R}$).