From 6373e3a8614a254560d76d45f10c8c8e098b79f3 Mon Sep 17 00:00:00 2001
From: ktzanev <kroum.tzanev@univ-lille.fr>
Date: Thu, 28 Dec 2023 18:55:33 +0100
Subject: [PATCH] ajustements dans le code

---
 TDs/M61B_2023-24_TD2.pdf | Bin 53856 -> 53859 bytes
 TDs/M61B_2023-24_TD2.tex |   6 +++---
 TDs/M61B_2023-24_TD3.pdf | Bin 65084 -> 65241 bytes
 TDs/M61B_2023-24_TD3.tex |  29 +++++++++++++++--------------
 TDs/M61B_2023-24_TD4.pdf | Bin 82865 -> 82756 bytes
 TDs/M61B_2023-24_TD4.tex |   6 +++---
 TDs/M61B_2023-24_TD5.tex |   4 ++--
 TDs/M61B_2023-24_TD6.pdf | Bin 77941 -> 77883 bytes
 TDs/M61B_2023-24_TD6.tex |  14 ++++++++------
 TDs/M61B_2023-24_TD7.pdf | Bin 80786 -> 80718 bytes
 TDs/M61B_2023-24_TD7.tex |  10 +++++-----
 TDs/M61B_2023-24_TD8.pdf | Bin 71412 -> 71764 bytes
 TDs/M61B_2023-24_TD8.tex |   2 +-
 13 files changed, 37 insertions(+), 34 deletions(-)

diff --git a/TDs/M61B_2023-24_TD2.pdf b/TDs/M61B_2023-24_TD2.pdf
index 8c42d0114b643cc9fd1f534bc956ffe60a156dc3..f174c0970ef6cc128d2166a9025a0e828db1bc02 100644
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diff --git a/TDs/M61B_2023-24_TD2.tex b/TDs/M61B_2023-24_TD2.tex
index 86c3b98..088da4e 100644
--- a/TDs/M61B_2023-24_TD2.tex
+++ b/TDs/M61B_2023-24_TD2.tex
@@ -28,7 +28,7 @@
 
   Soit $(A_n)_{n\in I}$ une partition d'un ensemble $E$ où $I\subset\N$.
   \begin{enumerate}
-    \item Caractériser les éléments de la tribu $\mathscr{T}\coloneqq\sigma(\{A_n: n\in I\})$ lorsque $I=\{0\}$, $I=\{0,1\}$, $I=\{0,1,2\}$, $I=\N$.
+    \item Caractériser les éléments de la tribu $\mathscr{T}\coloneqq\sigma(\ensemble{A_n}{n\in I})$ lorsque $I=\{0\}$, $I=\{0,1\}$, $I=\{0,1,2\}$, $I=\N$.
     \item Soit $f$ une fonction mesurable de $(E,\mathscr{T})$ dans $(\R,\mathcal{B}(\R))$. Montrer que $f$ est constante sur chaque $A_n$. En déduire la forme générale des applications mesurables pour $I$ comme dans la question précédente.
   \end{enumerate}
 \end{exo}
@@ -73,7 +73,7 @@
 
   Soient $f,g$ deux applications mesurables d'un espace mesurable $(E,\mathscr{T})$ à valeurs dans $\overline{\R}$ (muni de la tribu borélienne). Montrer que
   \[
-    \{x\in E:f(x)=g(x)\}\in \mathscr{T}.
+    \ensemble{x\in E}{f(x)=g(x)} \in \mathscr{T}.
   \]
   % \indication{On prendra garde au fait que les fonctions $f$ et $g$ peuvent prendre des valeurs infinies.}
 \end{exo}
@@ -87,7 +87,7 @@
     \item Supposons dans cette question que $\suiteN{f_n}$ converge simplement vers une application $f$. Montrer que $f$ est $(\mathscr{T},\mathcal{B}(E))$-mesurable.
     \item Supposons que $(F,d)$ soit complet. Montrer que
     \[
-      \{x\in E: \suiteN{f_n(x)} \mbox{ converge simplement}\}\in \mathscr{T}.
+      \ensemble{x\in E}{\suiteN{f_n(x)} \text{ converge simplement}} \in \mathscr{T}.
     \]
   \end{enumerate}
 \end{exo}
diff --git a/TDs/M61B_2023-24_TD3.pdf b/TDs/M61B_2023-24_TD3.pdf
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diff --git a/TDs/M61B_2023-24_TD3.tex b/TDs/M61B_2023-24_TD3.tex
index 4910ff8..c1542c8 100644
--- a/TDs/M61B_2023-24_TD3.tex
+++ b/TDs/M61B_2023-24_TD3.tex
@@ -12,7 +12,8 @@
 % ===============================================
 
 % -----------------------------------------------
-\begin{exo}\emph{La mesure de comptage}
+\begin{exo}\emph{(La mesure de comptage)}
+
   Soit $X$ un ensemble non vide. Pour $A\in \mathcal{P}(X)$, on pose
   \[
     c(A)\coloneqq\text{card}(A).
@@ -21,7 +22,7 @@
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{Montrer qu'une application est une mesure}
+\begin{exo}\emph{(Montrer qu'une application est une mesure)}
 
   Soit $(E, \mathscr{T})$ un espace mesurable. Dans chacun des cas suivants, montrer que l'application donnée est une mesure. %et décrire les ensembles de mesure nulle.
   \begin{enumerate}
@@ -32,7 +33,7 @@
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{Ensembles de mesure positive}
+\begin{exo}\emph{(Ensembles de mesure positive)}
 
   Soit $f:(E,\mathscr{T})\longrightarrow (\R,\mathcal{B}(\R))$ une fonction mesurable. Soit $\mu$ une mesure sur $(E,\mathscr{T})$.
   \begin{enumerate}
@@ -53,15 +54,15 @@
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{Régularité et mesure finie}
+\begin{exo}\emph{(Régularité et mesure finie)}
 
   Soit $\mu$ une mesure {\bf finie} sur $(\R,\mathcal{B}(\R))$. L'objectif de l'exercice est de montrer que pour tout $A\in\mathcal{B}(\R)$, on a
   \[
-    \mu(A)=\sup\{\mu(F): F \text{ fermé }, F\subset A\}
+    \mu(A)=\sup\ensemble{\mu(F)}{ F \text{ fermé }, F\subset A}
   \]
   et
   \[
-    \mu(A)=\inf\{\mu(O): O \text{ ouvert }, A\subset O\}.
+    \mu(A)=\inf\ensemble{\mu(O)}{ O \text{ ouvert }, A\subset O}.
   \]
   \begin{enumerate}
     \item Montrer qu'il suffit de prouver que tout borélien $A$ de $\R$ vérifie la propriété suivante :
@@ -71,11 +72,11 @@
     \[
       \text{ de $\R$ tels que $F\subset A\subset O$ et $\mu(O\setminus F)\leq \varepsilon$.}
     \]
-    \item Introduisons $\mathscr{T}=\{A\in\mathcal{B}(\R): A \text{ vérifie (P)}\}$.
+    \item Introduisons $\mathscr{T}=\ensemble{A\in\mathcal{B}(\R)}{ A \text{ vérifie (P)}}$.
     \begin{enumerate}
       \item Montrer que $\mathscr{T}$ contient $\mathcal{O}(\R)$, l'ensemble des ouverts de $\R$.
 
-      \emph{Indication :} soit $A\in\mathcal{O}(\R)$; on pourra considérer $F_{p}=\{x\in \R: d(x, A^c)\geq 1/p\}$, $p\geq 1$, et remarquer que $A=\bigcup_{p\geq 1} F_p$.
+      \emph{Indication :} soit $A\in\mathcal{O}(\R)$; on pourra considérer $F_{p}=\ensemble{x\in \R}{ d(x, A^c)\geq 1/p}$, $p\geq 1$, et remarquer que $A=\bigcup_{p\geq 1} F_p$.
       \item Montrer que $\mathscr{T}$ est stable par union dénombrable.
 
       \emph{Indication :} soit $A_n\in\mathscr{T}$, $n\geq 1$, et $\varepsilon>0$. Considérer alors $F_n$ fermé de $\R$ et $O_n$ ouvert de $\R$ tels que  $F_n\subset A_n\subset O_n$ et $\mu(O_n\setminus F_n)\leq \frac{\varepsilon}{2^{n+1}}$. Remarquer qu'il existe $n\geq 1$ tel que $\mu(\bigcup_{k\geq 1}F_n)\leq \mu(\bigcup_{k=1}^n F_k)+\frac{\varepsilon}{2}$.
@@ -122,13 +123,13 @@
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{Lebesgue nulle et intérieur}
+\begin{exo}\emph{(Lebesgue nulle et intérieur)}
 
   Montrer que tout sous-ensemble mesurable $E\subset \R^d$ de mesure de Lebesgue nulle est d'intérieur vide.
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{Un calcul avec la mesure de Lebesgue}
+\begin{exo}\emph{(Un calcul avec la mesure de Lebesgue)}
 
   On considère $\R$ muni de la tribu borélienne $\mathcal{B}(\R)$ et de la mesure de Lebesgue $\lambda$. Pour $n\geq 0$, on pose $A_n=]n,n+2^{-n}[$.
   \begin{enumerate}
@@ -139,11 +140,11 @@
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{Un deuxième calcul avec la mesure de Lebesgue}
+\begin{exo}\emph{(Un deuxième calcul avec la mesure de Lebesgue)}
 
   Soit $\suite[n\geq 1]{a_n})$ une suite de nombre réels. On pose
   \[
-    A=\{x\in\R: \exists n\in\N^*, |x-a_n|\leq 2^{-n}\}
+    A=\ensemble{x\in\R}{ \exists n\in\N^*, |x-a_n|\leq 2^{-n}}
   \]
   \begin{enumerate}
     \item Justifier que $A\in\mathcal{B}(\R)$.
@@ -155,7 +156,7 @@
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{Un ensemble non bor\'elien : l'ensemble de Vitali}
+\begin{exo}\emph{(Un ensemble non borélien : l'ensemble de Vitali)}
 
   Nous allons exhiber dans cet exercice une partie de $\R$ qui n'est pas dans la tribu des boréliens.
   \begin{enumerate}
@@ -171,7 +172,7 @@
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}\emph{Continuité et notion de presque partout}
+\begin{exo}\emph{(Continuité et notion de presque partout)}
 
   Soit $\lambda$ la mesure de Lebesgue sur $[0,1]$ (i.e. la restriction à $[0,1]$ de la mesure de Lebesgue). Soit $f:[0,1]\longrightarrow\R$ une fonction continue et supposons que $f$ est nulle sur un borélien de mesure $1$, c'est à dire que $\lambda\pa{f^{-1}(0)}=1$. Montrer alors que $f$ est identiquement nulle. Le résultat subsiste-t-il si on remplace continue par mesurable?
 \end{exo}
diff --git a/TDs/M61B_2023-24_TD4.pdf b/TDs/M61B_2023-24_TD4.pdf
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diff --git a/TDs/M61B_2023-24_TD4.tex b/TDs/M61B_2023-24_TD4.tex
index 14b14e3..8263ca5 100644
--- a/TDs/M61B_2023-24_TD4.tex
+++ b/TDs/M61B_2023-24_TD4.tex
@@ -218,9 +218,9 @@ Dans toute la fiche, on supposera que les variables aléatoires d'un même exerc
   \]
   L'événement $A_n$ est un retour à zéro. On pose
   \[
-    A\coloneqq\{\omega\in\Omega\; ;\;
-      \text{la suite $S_n(\omega)$ repasse une infinité de fois en $0$}\}=
-      \cap_{k\geq 1}\cup_{j\geq k}A_j.
+    A\coloneqq
+    \ensemble{\omega\in\Omega}{\text{la suite $S_n(\omega)$ repasse une infinité de fois en $0$}}
+    = \cap_{k\geq 1}\cup_{j\geq k}A_j.
   \]
   Prouver que $P(A)=0$.
 
diff --git a/TDs/M61B_2023-24_TD5.tex b/TDs/M61B_2023-24_TD5.tex
index c61193f..ec40612 100644
--- a/TDs/M61B_2023-24_TD5.tex
+++ b/TDs/M61B_2023-24_TD5.tex
@@ -120,8 +120,8 @@ sont définies sur un même espace de probabilités $(\Omega,\mathscr F, \P)$
       \quad \text{avec} \quad
     f(s) =
       \begin{cases}
-        0               &\mbox{ si }s<0\mbox{ ou }s \geq 10, \\
-        \lambda s(10-s) &\mbox{ si }0 \leq s < 10,
+        0               &\text{ si }s<0\text{ ou }s \geq 10, \\
+        \lambda s(10-s) &\text{ si }0 \leq s < 10,
       \end{cases}
   \]
   où $\lambda$ est un réel strictement positif.
diff --git a/TDs/M61B_2023-24_TD6.pdf b/TDs/M61B_2023-24_TD6.pdf
index 86246ad244631500db50547668f736fb9c28579b..6399ffd6aeb9b25191f895062f9e338ce4d4285e 100644
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diff --git a/TDs/M61B_2023-24_TD6.tex b/TDs/M61B_2023-24_TD6.tex
index c18e8c6..d03ee05 100644
--- a/TDs/M61B_2023-24_TD6.tex
+++ b/TDs/M61B_2023-24_TD6.tex
@@ -193,8 +193,8 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
   \[
     f_n(x) =
       \begin{cases}
-        g(x) \mbox{ si }n\mbox{ est pair},\\
-        g(-x) \mbox{ sinon.}
+        g(x) \text{ si }n\text{ est pair},\\
+        g(-x) \text{ sinon.}
       \end{cases}
   \]
   Montrer que
@@ -220,9 +220,9 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
   \[
     \lim_{n\to\infty}\int_\R n\log\pa{1+\pa{\frac{f(x)}{n}}^a}dx =
     \begin{cases}
-      \infty & \mbox{ si }0<a<1, \\
-      c      & \mbox{ si }a=1,   \\
-      0      & \mbox{ si }1<a
+      \infty & \text{ si }0<a<1, \\
+      c      & \text{ si }a=1,   \\
+      0      & \text{ si }1<a
     \end{cases}.
   \]
 \end{exo}
@@ -276,7 +276,9 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
 
   Supposons que $\mu$ est une  mesure finie, et $f:E\to\R$ une fonction mesurable. Pour tout $n\in\N$, on pose
   \[
-    A_n=\{x\in E, \abs{f(x)}\geq n\} \quad \mbox{ et }\quad B_n=\{x\in E, \;n\leq \abs{f(x)}< n+1\}.
+    A_n=\ensemble{x\in E}{n\leq\abs{f(x)}}
+    \quad \text{ et }\quad
+    B_n=\ensemble{x\in E}{n\leq \abs{f(x)}< n+1}.
   \]
   Montrer que les propositions suivantes sont équivalentes.
   \begin{enumerate}
diff --git a/TDs/M61B_2023-24_TD7.pdf b/TDs/M61B_2023-24_TD7.pdf
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diff --git a/TDs/M61B_2023-24_TD7.tex b/TDs/M61B_2023-24_TD7.tex
index 5613971..8b5b331 100644
--- a/TDs/M61B_2023-24_TD7.tex
+++ b/TDs/M61B_2023-24_TD7.tex
@@ -18,7 +18,7 @@
   \begin{enumerate}
     \item Montrer que $\mu$ est uniquement déterminée par la donnée de $F$.
     \item Montrer que $F$ est décroissante, continue à gauche sur $\R$ et calculer ses limites en $\pm \infty$.
-    \item  Calculer $\mu \{x\}$ pour $x\in \R$ et montrer que $F$ est continue en $x$ si et seulement si $\mu\{x\}=0$. Que peut-on en déduire sur $D=\{x\in \R, \ \mu\{x\}\not= 0\}$ ?
+    \item  Calculer $\mu \{x\}$ pour $x\in \R$ et montrer que $F$ est continue en $x$ si et seulement si $\mu\{x\}=0$. Que peut-on en déduire sur $D=\ensemble{x\in \R}{\mu\{x\}\not= 0}$ ?
   \end{enumerate}
 \end{exo}
 
@@ -44,8 +44,8 @@
     \[
       \delta_a(B) =
         \begin{cases}
-          1 \mbox{ si }a\in B\\
-          0 \mbox{ sinon.}
+          1 \text{ si }a\in B\\
+          0 \text{ sinon.}
         \end{cases}
     \]
     \begin{enumerate}
@@ -105,7 +105,7 @@
 
   Soit $F:\R\to\R$ une fonction croissante, continue à droite, vérifiant $ \lim_{-\infty}F=0$ et $\lim_{+\infty}F=1$. On veut démontrer qu'il existe une variable aléatoire $X$ dont $F$ est la fonction de répartition. Pour $u\in]0,1[$, on pose
   \[
-    G(u)=\inf\{x\in\R; F(x)\geq u\}.
+    G(u)=\inf\ensemble{x\in\R}{F(x)\geq u}.
   \]
   \begin{enumerate}
     \item Vérifier que $G$ est bien définie.
@@ -136,7 +136,7 @@
     \end{enumerate}
     \item On souhaite prouver que, parmi les variable aléatoires de variance donnée, les lois normales admettent une entropie maximale. On fixe $Y$ une variable aléatoire réelle centrée (c'est à dire d'espérance nulle), de densité $f_Y$ et de variance $\sigma^2$, admettant une entropie. On note $\varphi$ la densité d'une loi normale centrée ($m=0$), de variance $\sigma^2$. On suppose que les fonctions\vspace{-0.5em}
     \[
-      x\mapsto f_Y(x)\log\frac{\varphi(x)}{f_Y(x)} \quad \mbox{ et }\quad x\mapsto f_Y(x)\log\varphi(x)
+      x\mapsto f_Y(x)\log\frac{\varphi(x)}{f_Y(x)} \quad \text{ et }\quad x\mapsto f_Y(x)\log\varphi(x)
     \]
     sont intégrables sur $\R$.
     \begin{enumerate}
diff --git a/TDs/M61B_2023-24_TD8.pdf b/TDs/M61B_2023-24_TD8.pdf
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diff --git a/TDs/M61B_2023-24_TD8.tex b/TDs/M61B_2023-24_TD8.tex
index 21ed83d..17abbd0 100644
--- a/TDs/M61B_2023-24_TD8.tex
+++ b/TDs/M61B_2023-24_TD8.tex
@@ -59,7 +59,7 @@
     \[
       \PP{|S_n|\geq \lambda}\leq 2 \exp\pa{\frac{-\lambda^2}{2n}}
     \]
-    \item Soit $c>1$. Pour tout $n\in \N^*$, on définit l'événement $A_n=\{|S_n|\leq \sqrt{2cn \log n}\}$. En utilisant le Lemme de Borel Cantelli, montrer que
+    \item Soit $c>1$. Pour tout $n\in \N^*$, on définit l'événement $A_n=\Bigl\{|S_n|\leq \sqrt{2cn \log n}\Bigr\}$. En utilisant le Lemme de Borel Cantelli, montrer que
     \[
       \PP{\liminf A_n}=1.
     \]
-- 
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