Exercise 17, § 1

For every measure $\mu$ on a locally compact space $X$ and every homeomorphism $\sigma$ of $X$ onto itself, let $\mu_\sigma$ be the measure $f \mapsto \mu(f \circ \sigma)$.

a) Show that $|\mu_\sigma| = |\mu|_\sigma$.

b) Equip $\mathscr{M}(X; \mathbf{C})$ with the vague topology and denote by $\mathscr{A}$ the set of continuous endomorphisms of the locally convex space $\mathscr{M}(X; \mathbf{C})$; equip $\mathscr{A}$ with the topology of uniform convergence in the vaguely bounded subsets of $\mathscr{M}(X; \mathbf{C})$. Let $\Gamma$ be the group of homeomorphisms of X onto itself, equipped with the topology $\mathscr{T}_\beta$ defined in GT, X, §3, No. 5. For every $\sigma \in \Gamma$, let $A_\sigma$ be the mapping $\mu \mapsto \mu_\sigma$, which belongs to $\mathscr{A}$; show that the mapping $\sigma \mapsto A_\sigma$ of $\Gamma$ into $\mathscr{A}$ is continuous.