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 $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Show that there exists a sequence $(\varphi_k)_{k \in \mathbb{N}^*}$ of strictly increasing maps from $\mathbb{N}^*$ to $\mathbb{N}^*$ such that, for all $k \in \mathbb{N}^*$ and for all integer $1 \leqslant i \leqslant k$, the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ converges.
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 $E$.
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$.
Show that there exists a sequence $(\varphi_k)_{k \in \mathbb{N}^*}$ of strictly increasing maps from $\mathbb{N}^*$ to $\mathbb{N}^*$ such that, for all $k \in \mathbb{N}^*$ and for all integer $1 \leqslant i \leqslant k$, the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ converges.