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 $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and let $\mu$ be an element of $\mathscr{M}(E)$. Show that if the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in the normed vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$, then $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x).$$

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 $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
We fix a sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ and $\mu \in \mathscr{M}(E)$ satisfying $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x). \tag{1}$$ We also fix a real number $\varepsilon > 0$.
Show that there exists a finite subset $F_\varepsilon$ of $E$ and an integer $N_\varepsilon \geqslant 0$ such that $\mu(F_\varepsilon) > 1 - \varepsilon$ and for all integer $n \geqslant N_\varepsilon$ $$\sum_{x \in F_\varepsilon} |\mu_n(x) - \mu(x)| < \varepsilon.$$

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 $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
We fix a sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ and $\mu \in \mathscr{M}(E)$ satisfying condition (1). We also fix a real number $\varepsilon > 0$ and a finite subset $F_\varepsilon$ of $E$ and integer $N_\varepsilon \geqslant 0$ as in 10a.
Show that for every subset $A$ of $E$: $$|\mu_n(A) - \mu(A)| \leqslant |\mu_n(A \cap F_\varepsilon) - \mu(A \cap F_\varepsilon)| + \mu(E \backslash F_\varepsilon) + \mu_n(E \backslash F_\varepsilon)$$ and deduce that if $n \geqslant N_\varepsilon$, then $|\mu_n(A) - \mu(A)| < 4\varepsilon$.

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 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 let $\mu$ be an element of $\mathscr{M}(E)$. Show that if the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in the normed vector space $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$, then $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x)$$

Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$, such that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. We define the random variable: $$L : \Omega \longrightarrow \mathbb{N}, \quad \omega \longmapsto \begin{cases} 0 & \text{if } \forall n \in \mathbb{N}, X_n(\omega) = X(\omega) \\ \max\{n \in \mathbb{N}, X_n(\omega) \neq X(\omega)\} & \text{otherwise.} \end{cases}$$ Justify that the application $L$ is well defined.

We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote $$Q = \left\{(q_{ij})_{(i,j) \in I \times J} \in \mathbb{R}^{I \times J} \mid q_{ij} \geq 0 \text{ for all } (i,j) \in I \times J\right\}$$ and $$F(\alpha, \beta) = \left\{q \in Q \mid \sum_{j' \in J} q_{ij'} = \alpha_i \text{ and } \sum_{i' \in I} q_{i'j} = \beta_j \text{ for all } (i,j) \in I \times J\right\}.$$ Verify that $F(\alpha, \beta)$ is a convex set of the vector space $E = \mathbb{R}^{I \times J}$.

Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Show that the map $\varphi$ defined on $\left( L ^ { 0 } ( \Omega ) \right) ^ { 2 }$ by $$\forall X , Y \in L ^ { 0 } ( \Omega ) , \quad \varphi ( X , Y ) = \mathbf { E } ( X Y )$$ is an inner product on $L ^ { 0 } ( \Omega )$.

Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Let the map $\psi : u \in \mathbf { R } ^ { ( \mathbf { N } ) } \mapsto \sum _ { i = 0 } ^ { + \infty } u _ { i } X _ { i }$. Show that $\psi$ takes its values in $L ^ { 0 } ( \Omega )$, then that $\psi$ preserves the inner product.