Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $\mu_\infty$ as defined in 12b. We say that the sequence $(\mu_n)_{n \in \mathbb{N}}$ is tight if for all real $\varepsilon > 0$, there exists a finite subset $F_\varepsilon$ of $E$ such that $\mu_n(F_\varepsilon) \geqslant 1 - \varepsilon$ for all natural integer $n$. Suppose further that the sequence $(\mu_n)_{n \in \mathbb{N}}$ is tight. Show then that $\mu_\infty$ defines an element of $\mathscr{M}(E)$ which is a cluster point of the sequence $(\mu_n)_{n \in \mathbb{N}}$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $\mu_\infty$ as defined in 12b. We say that the sequence $(\mu_n)_{n \in \mathbb{N}}$ is tight if for all real $\varepsilon > 0$, there exists a finite subset $F_\varepsilon$ of $E$ such that $\mu_n(F_\varepsilon) \geqslant 1 - \varepsilon$ for all natural integer $n$. Suppose further that the sequence $(\mu_n)_{n \in \mathbb{N}}$ is tight. Show then that $\mu_\infty$ defines an element of $\mathscr{M}(E)$ which is a cluster point of the sequence $(\mu_n)_{n \in \mathbb{N}}$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.