Exercise 5, § 5

Let E be a real Hilbert space admitting an orthonormal basis $(x_n)_{n \in \mathbf{N}}$. Let T be the space E equipped with the weakened topology, and let $(a_n)_{n \in \mathbf{N}}$ be a sequence of real numbers such that $0 < a_n < 1$ for all n and $\lim_{n \to \infty} n^2 \log a_n = 0$. For every $n \in \mathbf{N}$, set $\mu_n = \sum_{p \in \mathbf{N}} a_n^p (1 - a_n) \cdot \varepsilon_{n \cdot x_p}$. Show that $\mu_n$ is a bounded positive measure on T of total mass 1 for every $n \in \mathbf{N}$, that $\lim_{n \to \infty} \mu_n = \varepsilon_0$ (tight convergence), and that $\lim_{n \to \infty} \mu_n(K) = 0$ for every compact subset K of T (apply the criterion of the preceding exercise to the set $\mathscr{U}$ of neighborhoods of 0 of the form $\{ x | (a|x| \leq 1 \}$, where a runs over E). In particular, the set of elements of the sequence $(\mu_n)_{n \in \mathbf{N}}$ is a relatively compact subset of $\mathscr{M}_+^b(T)$ that does not satisfy Prokhorov’s condition.