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 $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. For every integer $k \in \mathbb{N}^*$, we denote $\delta_k$ the probability measure on $E$ such that, for all $n \in \mathbb{N}^*$, $$\delta_k(\{x_n\}) = \begin{cases} 1 & \text{if } n = k \\ 0 & \text{otherwise} \end{cases}$$ Does the sequence $(\delta_k)_{k \in \mathbb{N}^*}$ converge 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 $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. For every integer $k \in \mathbb{N}^*$, we denote $\delta_k$ the probability measure on $E$ such that, for all $n \in \mathbb{N}^*$,
$$\delta_k(\{x_n\}) = \begin{cases} 1 & \text{if } n = k \\ 0 & \text{otherwise} \end{cases}$$
Does the sequence $(\delta_k)_{k \in \mathbb{N}^*}$ converge in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$?