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 $$\forall x \in E, \quad \lim_{n \rightarrow +\infty} \mu_n(x) = \mu(x)$$

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ and we denote by $\Omega \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined by
$$\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$$
where $(e _ { 1 } , \ldots , e _ { n })$ denotes the canonical basis of $\mathbb { R } ^ { n }$. Show that
$$\forall ( x , y ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } , \quad \omega ( x , y ) = X ^ { \top } \Omega Y$$
where $X$ and $Y$ denote the columns of the coordinates of $x$ and $y$ in the canonical basis of $\mathbb { R } ^ { n }$.

Deduce that $E$ is a vector subspace of the vector space $\mathcal { C } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { R } \right)$ of continuous functions on $\mathbb { R } _ { + } ^ { * }$ with values in $\mathbb { R }$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ 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 $E$. 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 $E$. 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 an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. 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): $\forall x \in E, \lim_{n \to +\infty} \mu_n(x) = \mu(x)$. We fix $\varepsilon > 0$ and a finite subset $F_\varepsilon$ of $E$ and integer $N_\varepsilon$ as in 10a. Show that for every subset $A$ of $E$: $$\left|\mu_n(A) - \mu(A)\right| \leqslant \left|\mu_n(A \cap F_\varepsilon) - \mu(A \cap F_\varepsilon)\right| + \mu(E \backslash F_\varepsilon) + \mu_n(E \backslash F_\varepsilon)$$ and deduce that if $n \geqslant N_\varepsilon$, then $\left|\mu_n(A) - \mu(A)\right| < 4\varepsilon$.

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 $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. 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 (1): $\forall x \in E, \lim_{n \to +\infty} \mu_n(x) = \mu(x)$.

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ with associated matrix $\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$, and that $\omega(x,y) = X^{\top} \Omega Y$ for all $x,y \in \mathbb{R}^n$. Deduce that $\Omega$ is antisymmetric and invertible.

For all functions $f \in E$ and $g \in E$, we set $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that this defines an inner product on $E$.

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
By considering a well-chosen sequence of functions, show that there does not exist an element $C$ of $\mathbf{R}^{+*}$ such that $$\forall f \in \mathcal{C}^{1}, \quad V(f) \leq C\|f\|_{\infty}$$

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 $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
For every integer $k \in \mathbb{N}^*$, we denote by $\delta_k$ 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 $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$?

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{12}$ ▷ Let $A \in \mathcal{A}_G$ and $B \in \mathcal{A}_G$. Show that the application $$\begin{aligned} u : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{R}) \\ t & \longmapsto u(t) = e^{tA} \cdot B \cdot e^{-tA} \end{aligned}$$ takes values in $\mathcal{A}_G$.

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
Show that the inner product $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ defined above depends only on the inner product on $E$ and not on the choice of the orthonormal basis $e$.

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ with associated matrix $\Omega$ that is antisymmetric and invertible. Conclude that the integer $n$ is even.

The norm $\| \cdot \|$ associated with the inner product $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ is defined for all functions $f \in E$ by $$\| f \| = \left( \int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t \right) ^ { 1 / 2 }$$ Show that $\lim _ { x \rightarrow 0 } \left\| k _ { x } \right\| = 0$. We recall that, for all $x > 0 , k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$.

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $f \in \mathcal{C}^{1}$ with real values. We assume that the set $C(f)$ of points in $]-\pi, \pi[$ where the function $f^{\prime}$ vanishes is finite. We denote by $\ell$ the cardinality of $C(f)$ and, if $\ell \geq 1$, we denote by $t_{1} < \cdots < t_{\ell}$ the elements of $C(f)$. We set $t_{0} = -\pi$ and $t_{\ell+1} = \pi$.
Show that $$V(f) = \sum_{j=0}^{\ell} \left|f\left(t_{j+1}\right) - f\left(t_{j}\right)\right|$$
For $0 \leq j \leq \ell$, let $\psi_{j}$ be the function from $\mathbf{R}$ to $\{0,1\}$ equal to 1 on $\left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$ and to 0 on $\mathbf{R} \backslash \left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$. Show that $$V(f) = \sum_{j=0}^{\ell} \int_{-\|f\|_{\infty}}^{\|f\|_{\infty}} \psi_{j}$$

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 $\mathscr{M}(E)$. Let $(\varphi_k)_{k \in \mathbb{N}^*}$ be the sequence of strictly increasing maps from 12a.
Show that for all $i \in \mathbb{N}^*$ and all integer $k \geqslant i$, the limit of the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ depends only on $i$ and not on $k$. We denote this limit by $\mu_\infty(x_i)$.

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 $\mathscr{M}(E)$. Let $(\varphi_k)_{k \in \mathbb{N}^*}$ be the sequence of strictly increasing maps from 12a, and let $\mu_\infty(x_i)$ be the limits defined in 12b.
Show that the map $$\begin{array}{rcl} \psi : \mathbb{N}^* & \longrightarrow & \mathbb{N}^* \\ k & \longmapsto & \varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(k) \end{array}$$ is strictly increasing, and that, for all integer $i \in \mathbb{N}^*$, the sequence $\left(\mu_{\psi(k)}(x_i)\right)_{k \in \mathbb{N}^*}$ converges to $\mu_\infty(x_i)$.

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 $\mathscr{M}(E)$. Let $\mu_\infty(x_i)$ be the limits defined in 12b.
Show that $\mu_\infty(x_i) \geqslant 0$ for all $i$ in $\mathbb{N}^*$, and that $\sum_{i=1}^{\infty} \mu_\infty(x_i) \leqslant 1$.

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 $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Let $\mu_\infty$ be defined as in 12b and 12d.
We say that the sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ is tight if for every real number $\varepsilon > 0$, 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 $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Show that there exists a sequence $(\varphi_k)_{k \in \mathbb{N}^*}$ of strictly increasing applications 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.