Exercise 21, § 4

Let $G$ be a locally compact group, $\beta$ a left Haar measure on $G$; one canonically identifies $L^1(G, \beta)$ with a subspace of $\mathscr{M}^1(G)$. a) With the notations of §3, Exer. 11, let $A$ be a relatively compact subset of $\mathscr{M}_{III}$, $B$ a subset of $L^1(G, \beta)$, relatively compact in $\mathscr{M}_{IV}$. Show that the restriction to $A \times B$ of the mapping $(\mu, \nu) \mapsto \mu * \nu$ of $\mathscr{M}_{III} \times \mathscr{M}_{IV}$ into $\mathscr{M}_{IV}$ is continuous. (First prove that if $A$ and $B$ are compact, the image of $A \times B$ under this mapping is compact; use Exer. 10 of §3, as well as the criterion $\alpha$) of Ch. V, §5, Exer. 15. To show next that $(\mu, \nu) \mapsto \mu * \nu$ is continuous, make use of Exer. 11 c) of §3.) b) Let $\mathscr{U}_s^\infty(G)$ be the set of bounded numerical functions on $G$ uniformly continuous for the left uniform structure. Denote by $\mathscr{T}_{II}$ the topology $\sigma(\mathscr{M}^1(G), \mathscr{U}_s^\infty(G))$, and by $\mathscr{M}_{II}$ the space $\mathscr{M}^1(G)$ equipped with $\mathscr{T}_{II}$. Taking $G = \mathbf{R}$, give an example of a sequence $(\mu_n)$ of measures on $G$ tending to 0 for $\mathscr{T}_{II}$ and a sequence $(f_n)$ of functions in $L^1(G, \beta)$, tending to 0 for $\mathscr{T}_{IV}$ (or, what comes to the same, for the topology $\sigma(L^1(G, \beta), L^\infty(G, \beta))$), such that the sequence $(\mu_n * f_n)$ does not tend to 0 for $\mathscr{T}_{IV}$ (take $f_n(t) = \sin nt$ in the interval $[0, \pi]$, $f_n(t) = 0$ elsewhere). c) Let $A$ be a relatively compact subset of $\mathscr{M}_{II}$, $B$ a subset of $L^1(G, \beta)$, relatively compact for the topology of the norm $\|f\|_1$. Show that the restriction to $A \times B$ of the mapping $(\mu, f) \mapsto \mu * f$ of $\mathscr{M}_{III} \times L^1(G, \beta)$ into $L^1(G, \beta)$ is continuous ($L^1(G, \beta)$ being equipped with its normed space topology). (Reduce to proving that for $f \in \mathscr{H}(G)$, the mapping $\mu \mapsto \mu * f$ of $\mathscr{M}_{II}$ into $L^1(G, \beta)$ is continuous in $A$. Using the fact that for $g \in L^\infty(G, \beta)$ and $f \in \mathscr{H}(G)$, $g * f$ is uniformly continuous for the left uniform structure, first show that the mapping $\mu \mapsto \mu * f$ of $\mathscr{M}_{II}$ into $\mathscr{M}_{IV}$ is continuous. Next, using Exer. 20 a), reduce to proving that if $C$ is a subset of $L^1(G, \beta)$ compact for the topology

(*) For an example where $\mu$ is not a point measure and is such that $\mu * f \geq 0$ implies $f \geq 0$, see J.H. Williamson, Proc. Edinburgh Math. Soc., 11 (1958/59), 71–77.

σ(L^1(G, β), L^∞(G, β)) and h ∈ $\mathscr{K}(G)$, the mapping μ ↦ μ \* h of C into L^1(G, β) is continuous when L^1(G, β) is equipped with its normed space topology. By the same reasoning as in a), show that for this it suffices to prove that the image of C under this mapping is compact for the normed space topology. To this end, use Šmulian’s theorem, the fact that C is cramped (§3, Exer. 10) and Lebesgue’s theorem.)

d) Let A be a relatively compact subset of $\mathscr{P}_{II}$ containing 0, B a relatively compact subset of $\mathscr{P}_I$; show that for every ν_0 ∈ B, the restriction to A × B of the mapping (μ, ν) ↦ μ \* ν of $\mathscr{P}_{II}$ × $\mathscr{P}_I$ into $\mathscr{P}_{II}$ is continuous at the point (0, ν_0). (Make use of Exer. 20 a), and Exer. 21 c).)

e) Let A be a bounded subset of $\mathscr{P}^1$(G), f a function in L^1(G, β); show that the restriction to A of the mapping μ ↦ μ \* f of $\mathscr{P}_{III}$ into $\mathscr{P}_{IV}$ is continuous.

f) Let $\mathscr{P}^1_+$(G) be the set of bounded positive measures on G. Show that if a subset A of $\mathscr{P}^1_+$(G) is compact for $\mathscr{P}_{II}$, it is also compact for $\mathscr{P}_{III}$. (Reduce to proving that A is a cramped set. Argue by contradiction, using the fact that for f ∈ $\mathscr{K}(G)$, the image of A under the mapping μ ↦ μ \* f is a cramped set by virtue of c).)

g) Show that for G = $\mathbf{R}$, the topologies induced on $\mathscr{P}^1_+$(G) by $\mathscr{P}_{II}$ and $\mathscr{P}_{III}$ are distinct.