Exercise 6, § 1

Let $f$ be a numerical function $\geq 0$ defined on X.

a) Show that, for the mapping $\mu \mapsto \mu^*(f)$ of $\mathscr{M}_+(X)$ into $\overline{\mathbf{R}}$ to be continuous for the vague topology, it is necessary (and sufficient) that $f$ be continuous with compact support (make use of Exer. 2). For $\mu \mapsto \mu^*(f)$ to be lower semi-continuous for the vague topology, it is necessary (and sufficient, cf. Prop. 4) that $f$ be lower semi-continuous.

b) Show that, for the mapping $\mu \mapsto \mu^*(f)$ of $\mathscr{M}_+(X)$ into $\overline{\mathbf{R}}$ to be continuous for the quasi-strong topology (Ch. III, §1, Exer. 8), it is necessary and sufficient that $f$ be bounded and have compact support (method analogous to that of a)). From this, deduce that for every function $f \geq 0$ that is zero on the complement of a countable union of compact sets, the mapping $\mu \mapsto \mu^*(f)$ is lower semi-continuous for the quasi-strong topology (make use of Th. 3) (cf. Exer. 7 b)).

c) Show that, for the mapping $\mu \mapsto \mu^*(f)$ of $\mathscr{M}_+(X) \cap \mathscr{M}^1(X)$ into $\overline{\mathbf{R}}$ to be continuous for the ultrastrong topology (Ch. III, §1, Exer. 15), it is necessary and sufficient that $f$ be bounded; for every function $f \geq 0$ defined on X, $\mu \mapsto \mu^*(f)$ is lower semi-continuous for the ultrastrong topology.

d) Show that, for the mapping $\mu \mapsto \mu^*(f)$ of $\mathscr{M}_+(X) \cap \mathscr{M}^1(X)$ into $\overline{\mathbf{R}}$ to be continuous for the weak topology (Ch. III, §1, Exer. 15), it is necessary and sufficient that $f$ be continuous on X and tend to 0 at the point at infinity; for $\mu \mapsto \mu^*(f)$ to be lower semi-continuous for the weak topology, it is necessary and sufficient that $f$ be lower semi-continuous.