Exercise 21, § 3

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, exerc. 11, let A be a relatively compact subset of $\mathscr{M}_{\text{III}}$, B a subset of $L^1(G, \beta)$, relatively compact in $\mathscr{M}_{\text{IV}}$. Now prove that the restriction to $A \times B$ of the mapping $(\mu, v) \to \mu * v$ from $\mathscr{M}_{\text{III}} \times \mathscr{M}_{\text{IV}}$ into $\mathscr{M}_{\text{IV}}$ is continuous. (Proving first that if A and B are compact, the image of $A \times B$ by this mapping is compact; use exerc. 10 of § 3, as well as the $\alpha$ criterion) of Chap. V, § 5, exerc. 17. To show afterwards that $(\mu, v) \to \mu * v$ is continuous, use exerc. 11 c) of § 3.)

b) Let $\mathscr{U}_s^\infty(G)$ be the set of bounded numerical functions and uniformly continuous in G for the left uniform structure. Let $\mathscr{T}_{\text{II}}$ denote the topology $\sigma(\mathscr{M}^1(G), \mathscr{U}_s^\infty(G))$, and $\mathscr{M}_{\text{II}}$ the space $\mathscr{M}^1(G)$ endowed with $\mathscr{T}_{\text{II}}$. Taking $G = \mathbf{R}$, give an example of a sequence $(\mu_n)$ of measures on G tending to 0 for $\mathscr{T}_{\text{II}}$ and of a sequence $(f_n)$ of functions of $L^1(G, \beta)$, tending to 0 for $\mathscr{T}_{\text{IV}}$ (or, equivalently, 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 $\mathscr{T}_{\text{IV}}$ (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 $\mathscr{M}_{\text{II}}$, B a subset of $L^1(G, \beta)$, relatively compact for the topology of the norm $\|f\|_1$. Now prove that the restriction to $A \times B$ of the mapping $(\mu, f) \to \mu * f$ from $\mathscr{M}_{\text{III}} \times L^1(G, \beta)$ into $L^1(G, \beta)$ is continuous ($L^1(G, \beta)$ being endowed with its normed space topology). (Reduce to proving that for $f \in \mathscr{K}(G)$, the mapping $\mu \to \mu * f$ from $\mathscr{M}_{\text{II}}$ into $L^1(G, \beta)$ is continuous in A. Using the fact that for $g \in L^\infty(G, \beta)$ and $f \in \mathscr{K}(G)$, $g * f$ is uniformly continuous for the left uniform structure, first show that the mapping $\mu \to \mu * f$ from $\mathscr{M}_{\text{II}}$ into $\mathscr{M}_{\text{IV}}$ is continuous. Using then exerc. 20 a), reduce to proving that if C is a subset of $L^1(G, \beta)$ compact for the topology $\sigma(L^1(G, \beta), L^\infty(G, \beta))$ and $h \in \mathscr{K}(G)$, the mapping $\mu \to \mu * h$ from C into $L^1(G, \beta)$ is continuous when $L^1(G, \beta)$ is endowed with its normed space topology. By the same reasoning as in a), show that it suffices for this to prove that the image of C by this mapping is compact for the normed space topology. Use for this the th. of Šmulian, the fact that C is tight (§ 3, exerc. 10) and the th. of Lebesgue).

d) Let A be a relatively compact subset of $\mathscr{M}_{II}$ containing 0, B a relatively compact subset of $\mathscr{M}_I$; now prove that for every $v_0 \in B$ the restriction to $A \times B$ of the mapping $(\mu, \nu) \to \mu * \nu$ from $\mathscr{M}_{II} \times \mathscr{M}_I$ into $\mathscr{M}_{II}$ is continuous at the point $(0, v_0)$. (Use exerc. 20 a), and exerc. 8 c.)

e) Let A be a bounded subset of $\mathscr{M}^1(G)$, $f$ a function of $L^1(G, \beta)$; now prove that the restriction to A of the mapping $\mu \to \mu * f$ from $\mathscr{M}_{III}$ into $\mathscr{M}_{IV}$ is continuous.

f) Let $\mathscr{M}_+^1(G)$ be the set of bounded positive measures on G. Prove that if a subset A of $\mathscr{M}_+^1(G)$ is compact for $\mathscr{T}_{II}$ it is also compact for $\mathscr{T}_{III}$. (Reduce to proving that A is a tight set. Argue by contradiction, using the fact that for $f \in \mathscr{K}(G)$, the image of A by the mapping $\mu \to \mu * f$ is a tight set by virtue of c.).

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