Exercise 8, § 5

Let T be a complete separable metric space, and d its metric. For every closed subset F of T and every real number $a > 0$, denote by $F^a$ the set of $x \in T$ such that $d(x, F) < a$. Given two bounded positive measures $\lambda$ and $\mu$ on T, denote by D($\lambda, \mu$) the infimum of the set of real numbers $a > 0$ satisfying the inequalities $\lambda^\bullet(F) \leq \mu^\bullet(F^a) + a$, $\mu^\bullet(F) \leq \lambda^\bullet(F^a) + a$ for every closed subset F of T.

a) Show that D is a metric on $\mathscr{M}_+^b$ and that $D(\varepsilon_x, \varepsilon_y) = d(x, y)$ for two points x and y of T such that $d(x, y) < 1$.

b) Let f be a Borel mapping of T into T (cf. TG, IX, §6, No. 3 or GT, IX, §6, Exer. 16) and $\lambda$ a bounded positive measure on T. Show that there exists a bounded positive measure $\mu$ on T such that $\mu^\bullet(A) = \lambda^\bullet(f^{-1}(A))$ for every Borel subset A of T. Let $a > 0$, and let H be the set of $x \in T$ such that $d(x, f(x)) \geq a$; show that H is Borel in T and that $D(\lambda, \mu) \leq \sup(a, \lambda(H))$ (for every closed subset F of T, one has $F \cap (T - H) \subset f^{-1}(F^a)$ and $\lambda^\bullet(F) \leq \lambda^\bullet(F \cap (T - H)) + \lambda^\bullet(H)$).

c) Let $\mu$ and $\mu_n$ (for $n \geq 1$) be bounded positive measures on T such that $\lim_{n \to \infty} D(\mu_n, \mu) = 0$. Show that the sequence $(\mu_n)$ converges tightly to $\mu$ (show that $\lim_{n \to \infty} \mu_n^\bullet(T) = \mu^\bullet(T)$ and $\mu^\bullet(F) \geq \limsup_{n \to \infty} \mu_n^\bullet(F)$ for every closed subset F of T; deduce from this that $\mu^\bullet(f) \leq \liminf_{n \to \infty} \mu_n^\bullet(f)$ for every lower semi-continuous function $f \geq 0$ by the method of Lemma 3 of §2, No. 6).

d) Conversely, show that for every sequence of bounded positive measures $\mu_n$ tending tightly to $\mu$ in $\mathscr{M}_+^b$, one has $\lim D(\mu_n, \mu) = 0$. One can proceed as follows: let $\varepsilon > 0$, and let K be a compact subset of T such that $\mu^\bullet(T - K) < \varepsilon$ and $\sup_{n} \mu_n^\bullet(T - K) < \varepsilon$; construct a finite family $(B_i)_{1 \leq i \leq p}$ of sets with $\mu$-negligible boundary, of diameter $\leq \varepsilon/2$, pairwise disjoint, whose union contains K (cf. Exer. 1). Let f be a mapping of T into T, constant on each of the sets $B_1, \ldots, B_p$ and $\mathscr{C}(B_1 \cup \cdots \cup B_p)$ and such that $f(B_i) \subset B_i$ for $1 \leq i \leq p$. Define the measures $\pi_n$ and $\pi$ by $\pi_n^\bullet(A) = \mu_n^\bullet(f^{-1}(A))$ and $\pi^\bullet(A) = \mu^\bullet(f^{-1}(A))$ for every Borel subset A of T; deduce from b) the relations $D(\pi_n, \mu_n) \leq \varepsilon$, $D(\pi, \mu) \leq \varepsilon$ and show that $\lim_{n \to \infty} D(\pi_n, \pi) = 0$.

e) The metric D on $\mathscr{M}_+^b$ is compatible with the tight topology (observe that $\mathscr{M}_+^b$ is metrizable for the tight topology).