Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. 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 $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$. Deduce that the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ if and only if it satisfies condition $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x). \tag{1}$$
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. 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 $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Deduce that the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ if and only if it satisfies condition
$$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x). \tag{1}$$