Exercise 9, § 5

Notations and hypotheses are those of the preceding exercise. Let $(\mu_n)$ be a Cauchy sequence in $\mathscr{M}_+^b$ for the metric D; show that the sequence $(\mu_n)$ is convergent (this gives a new proof of the fact that the space $\mathscr{M}_+^b$ is Polish for the tight topology). One can proceed thus:

a) Let $\varepsilon > 0$ and $a > 0$ be two real numbers; there exists a finite subset F of T such that $\sup_n \mu_n^\bullet(T - F^a) \leq \varepsilon$ (choose an integer $N \geq 1$ and a compact subset K of T such that $\sup_{n \geq N} D(\mu_n, \mu_N) \leq \varepsilon/2$ and $\mu_N^\bullet(T - K) \leq \varepsilon/2$; deduce from this that $\sup_{n \geq N} |\mu_n^\bullet(T) - \mu^\bullet(T)| \leq \varepsilon$ and $\sup_{n \geq N} \mu_n^\bullet(T - K^{a/2}) \leq \varepsilon$; finally, choose a finite subset F of T such that $K^{a/2} \subset F^a$ and $\sup_{n < N} \mu_n^\bullet(T - F^a) \leq \varepsilon$.

b) Let $\varepsilon > 0$; there exists a compact subset K of T such that $\sup \mu_n^\bullet(T - K) \leq \varepsilon$ (for every integer $p \geq 1$ choose a finite subset $F_p$ of T such that $\mu_n^\bullet(T - (F_p)^{2^{-p}}) \leq \varepsilon/2^p$, and set $K = \bigcap_{p \geq 1} (F_p)^{2^{-p}}$).

c) Deduce from b) that the Cauchy sequence $(\mu_n)$ has a convergent subsequence (for the metric D).