From b8ebdfb9a677dbba8c4ce1cef4f9af6baafdb84d Mon Sep 17 00:00:00 2001
From: ktzanev <kroum.tzanev@univ-lille.fr>
Date: Thu, 7 Mar 2024 10:54:28 +0100
Subject: [PATCH] ajustement de la fiche 6 avant impression

---
 TDs/M61B_2023-24_TD6.pdf | Bin 77883 -> 73791 bytes
 TDs/M61B_2023-24_TD6.tex | 133 +++++++++++----------------------------
 2 files changed, 37 insertions(+), 96 deletions(-)

diff --git a/TDs/M61B_2023-24_TD6.pdf b/TDs/M61B_2023-24_TD6.pdf
index 6399ffd6aeb9b25191f895062f9e338ce4d4285e..93ba84c982e6a0c5046d25367b97386c3c580d38 100644
GIT binary patch
delta 45557
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diff --git a/TDs/M61B_2023-24_TD6.tex b/TDs/M61B_2023-24_TD6.tex
index d03ee05..5eeb9da 100644
--- a/TDs/M61B_2023-24_TD6.tex
+++ b/TDs/M61B_2023-24_TD6.tex
@@ -6,24 +6,24 @@
 
 \begin{document}
 
-Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur un espace mesuré $(E, \mathscr{T}, \mu)$.
+Dans toute la fiche, on supposera, sauf mention contraire, que l'on se trouve sur un espace mesuré $(E, \mathscr{T}, \mu)$.
 
 
 % ===============================================
-\section{intégrale de Lebesgue}
+\section{Intégrale de Lebesgue}
 % ===============================================
 
 % -----------------------------------------------
 \begin{exo}\emph{(Quizz)}
   \begin{enumerate}
     \item La somme de deux fonctions intégrables est elle intégrable ?
-    \item Le carré d'une fonction intégrable est il intégrable ? Une fonction de carré intégrable est-elle elle même intégrable ?
+    \item Le carré d'une fonction intégrable est il intégrable ? Une fonction de carré intégrable est-elle elle-même intégrable ?
     \item La composée de deux fonctions intégrable est-elle intégrable ?
   \end{enumerate}
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Calcul d'une intégrale)}
 
   On considère la fonction $f:[0,1]\to\R$ définie par
   \[
@@ -41,18 +41,18 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
 % ===============================================
 
 % -----------------------------------------------
-\begin{exo}[.7]
+\begin{exo}[.49]\emph{(Inversion intégrale et limite ?)}
 
-  On note $\lambda$ la mesure de Lebesgue sur $\R$. Soit $f_n=\mathds{1}_{[n,2n]}$. Vérifier que les $f_n$ sont postives et mesurables. Calculer $\lim_{n\to +\infty} \int_\R f_nd\lambda$ et $\int_\R\lim_{n\to +\infty} f_nd\lambda$.
+  On note $\lambda$ la mesure de Lebesgue sur $\R$. Soit $f_n=\1_{[n,2n]}$. Vérifier que les $f_n$ sont positives et mesurables. Calculer $\lim_{n\to +\infty} \int_\R f_nd\lambda$ et $\int_\R\lim_{n\to +\infty} f_nd\lambda$.
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Calcul d'une intégrale limite)}
 
-  On pose $I(\alpha)=\lim_{n\to +\infty} \int_0^n\left(1-\frac xn\right)^ne^{\alpha x}dx$ pour $n\in\N$ et $\alpha\in\R$.
+  On pose $I_n(\alpha)=\int_0^n\left(1-\frac xn\right)^ne^{\alpha x}dx$ pour $n\in\N^{*}$ et $\alpha\in\R$.
   \begin{enumerate}
-    \item On définit pour $n\in\N$, $f_n:\R^+\to\R$ par $f_n(x)=\left(1-\frac xn\right)^n e^{\alpha x}\mathds{1}_{[0,n]}(x)$. En étudiant $g_n(x)=(n+1)\log\left(1-\frac{x}{n+1}\right)-n\log\left(1-\frac{x}{n}\right)$, montrer que $\suite{f_n}$ est une suite décroissante de fonctions.
-    \item En déduire la valeur de $I(\alpha)$ en fonction de $\alpha$.
+    \item On définit pour $n\in\N^{*}$, $f_n:\R_+\to\R$ par $f_n(x)=\left(1-\frac xn\right)^n e^{\alpha x}\1_{[0,n]}(x)$. En étudiant $g_n(x)=(n+1)\log\left(1-\frac{x}{n+1}\right)-n\log\left(1-\frac{x}{n}\right)$, montrer que $\suite{f_n}$ est une suite croissante de fonctions.
+    \item En déduire que $I(\alpha) \coloneqq \lim_{n\to +\infty} I_{n(\alpha)}$ existe et calculer sa valeur en fonction de $\alpha$.
   \end{enumerate}
 \end{exo}
 
@@ -64,9 +64,9 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
     f_n(x)=\frac1x\1_{[n,+\infty[}(x).
   \]
   \begin{enumerate}
-    \item Les fonctions $f_n$ sont elles intégrables sur $\R$ ?
-    \item Dans le théorème de convergence monotone, peut on remplacer \enquote{croissante} par \enquote{monotone} ?
-    \item Montrer que si l'on suppose qu'il existe $n_0$ tel que $\int_E f_{n_0}d\mu<\infty$, le théorème de convergence monotone s'applique également aux suites décroissantes de fonctions positives.
+    \item Les fonctions $f_n$ sont-elles intégrables sur $\R$ ?
+    \item Dans le théorème de convergence monotone, peut-on remplacer \enquote{croissante} par \enquote{monotone} ?
+    \item Soit $\suite{g_n}$ est une suite décroissante de fonctions réelles sur un espace mesuré. Montrer que s'il existe $n_0$ tel que $g_{n_0}$ soit intégrable, la conclusion du théorème de convergence monotone s'applique.
   \end{enumerate}
 \end{exo}
 
@@ -80,7 +80,7 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
         \sum_{n=0}^{+\infty} \int_Ef_nd\mu=\int_E\left(\sum_{n=0}^{+\infty} f_n\right)d\mu.
       \]
     \item En déduire la valeur de $\sum_{n=3}^{+\infty}\int_1^{+\infty}\frac x{(1+x)^n}dx.$
-    \item  Pour tout entier $n\geq 1$ et tout $x\in\R$, soit $f_n(x)=e^{-nx}-2e^{-2nx}$.
+    \item Pour tout entier $n\geq 1$ et tout $x\in\R$, soit $f_n(x)=e^{-nx}-2e^{-2nx}$.
     \begin{enumerate}
       \item Montrer que $\sum_{n\geq 1} f_n(x)$ est une série convergente pour tout $x>0$ et calculer sa somme $f(x)$.
       \item Comparer $\int_0^{+\infty} f(x)dx$ et $\sum_{n=1}^{+\infty} \int_0^{+\infty} f_n(x)dx$. Expliquer.
@@ -91,23 +91,23 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
 % -----------------------------------------------
 \begin{exo}\emph{(Mesure de comptage)}
 
-  On rappelle que la mersure de comptage $\mu$ est définie sur $\pa{\N,\mathcal{P}(\N)}$ par $\mu(A)=\mathrm{card}(A)$ si $A$ est fini et $\mu(A)=+\infty$ sinon.
+  On rappelle que la mesure de comptage $\mu$ est définie sur $\pa{\N,\mathcal{P}(\N)}$ par $\mu(A)=\mathrm{card}(A)$ si $A$ est fini et $\mu(A)=+\infty$ sinon.
   \begin{enumerate}
-    \item Soit $n\in\N$. Calculer $\int_{\N} \mathds{1}{\{n\}}d\mu$,  $\int_{\N} \sum_{k=0}^n\frac1{2^k}\mathds{1}{\{k\}}d\mu$
+    \item Soit $n\in\N$. Calculer $\int_{\N} \1_{\{n\}}d\mu$, $\int_{\N} \sum_{k=0}^n\frac1{2^k}\1_{\{k\}}d\mu$
     \item Soit $f:\N\to \R$ définie par $f(n)=\frac1{2^n}$. Calculer $\int_{\N} fd\mu$.
-    \item Soit $f:\N \to \R$ positive. Justifier que $\int_{\N}fd\mu=\sum_{n=0}^{+\infty} f(n)$.
-    \item Soit $\suite[n,p\in\N]{u_{n,p}}$ une suite de réels positifs. Démontrer que
-      \[
-        \sum_{n=0}^{+\infty} \sum_{p=0}^{+\infty} u_{n,p}=\sum_{p=0}^{+\infty} \sum_{n=0}^{+\infty} u_{n,p}.
-      \]
-    \item En déduire la valeur de $\sum_{p=2}^{+\infty} \sum_{n=2}^{+\infty} \frac1{n^p}$.
+    \item Soit $f:\N \to \R_{+}$. Justifier que $\int_{\N}fd\mu=\sum_{n=0}^{+\infty} f(n)$.
+    \item Soit $f:\N \to \R$. Montrer que $f$ est $\mu$-intégrable si et seulement si $\sum f(n)$ est absolument convergente et que, dans ce cas, $\int_{\N}fd\mu=\sum_{n=0}^{+\infty} f(n)$.
+    \item Donner l'exemple d'une fonction $f:\N \to \R$ telle que $\int_{\N}fd\mu$ n'est pas bien défini, mais telle que $\sum_{n=0}^{+\infty} f(n) \in \R$ est bien définie.
   \end{enumerate}
 \end{exo}
 
 % -----------------------------------------------
 \begin{exo}\emph{(Majoration d'intégrales qui passe à la limite)}
 
-  Soit $\suiteN{f_n}$ une suite de fonctions positives convergeant simplement vers $f$. On suppose qu'il existe une constante $K$ telle que $\int f_n d\mu\leq K$ pour tout $n$. Montrer que $\int fd\mu\leq K$.
+  Soit $\suiteN{f_n}$ une suite de fonctions positives convergeant simplement vers $f$. On suppose qu'il existe une constante $K$ telle que $\int f_n d\mu\leq K$ pour tout $n$. Montrer que $\int fd\mu\leq K$.\\
+  \begin{indication}
+    Considérer la suite $g_{n} = \inf_{k\geq n} f_{k}$.
+  \end{indication}
 \end{exo}
 
 
@@ -116,7 +116,7 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
 % ===============================================
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Limites d'intégrales)}
 
  Calculer les limites suivantes :
  \begin{enumerate}
@@ -134,7 +134,7 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Limites de séries)}
 
  En ré-écrivant les séries comme des intégrales par rapport à la mesure de comptage, calculer
  \begin{enumerate}
@@ -144,18 +144,19 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Intégration sur des \enquote{petits} ensembles)}
 
   Soit $(X,\mathcal{B},\mu)$ un espace mesuré et soit $f$ une fonction intégrable.
   \begin{enumerate}
-    \item Soit $\suite[n\geq1]{A_n}$ une suite d'ensembles mesurables tels que pour tout $n\in\N^*$, $\mu(A_n)<\frac1{n^2}$.  On pose $f_n=f\mathds{1}_{A_n}$.\\ Montrer que $\lim_{n\to +\infty} \mu\pa{\cup_{k=n}^{+\infty} A_k}=0$. En déduire que pour tout $\mu$-presque $x$, la suite $\suite{f_n(x)}$ converge vers $0$.
+    \item Soit $\suite[n\geq1]{A_n}$ une suite d'ensembles mesurables tels que pour tout $n\in\N^*$, $\mu(A_n)<\frac1{n^2}$. On pose $f_n=\abs{f}\1_{A_n}$.\\ Montrer que $\lim_{n\to +\infty} \mu\pa{\cup_{k=n}^{+\infty} A_k}=0$. En déduire que pour $\mu$-presque tout $x$, la suite $\suite{f_n(x)}$ est nulle à partir d'un certain rang.
     \item Montrer que $\lim_{n\to +\infty}\int_E f_n d\mu =0$.
-    \item En déduire que pour tout $\varepsilon>0$, il existe $\delta>0$ tel que pour tout $A\in\mathcal{B}$,  $\mu(A)<\delta$ implique $\int_A\abs{f} d\mu<\varepsilon$.
+    \item En déduire que pour tout $\varepsilon>0$, il existe $\delta>0$ tel que pour tout $A\in\mathcal{B}$, $\mu(A)<\delta$ implique $\int_A\abs{f} d\mu<\varepsilon$.
   \end{enumerate}
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Série d'intégrales)}
+
   Soit $\suiteN{f_n}$ une suite de $\mathcal{L}^1(\mu)$. On suppose que
   \[
     \sum_{n=1}^\infty\int_E |f_n|d\mu<\infty.
@@ -179,7 +180,7 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
 % -----------------------------------------------
 \begin{exo}\emph{(Inégalité de Fatou stricte I)}
 
-  On note $\lambda$ la mesure de Lebesgue. Soit $f_n=n\mathds{1}_{\left[0,\frac1n\right]}$.
+  On note $\lambda$ la mesure de Lebesgue. Soit $f_n=n\1_{\left[0,\frac1n\right]}$.
   \begin{enumerate}
     \item Appliquer si possible le lemme de Fatou.
     \item Calculer $\liminf_{n\to +\infty} \int_{\R} f_nd\lambda$ et $\int_{\R} \liminf_{n\to +\infty} f_nd\lambda$
@@ -189,7 +190,7 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
 % -----------------------------------------------
 \begin{exo}\emph{(Inégalité de Fatou stricte II)}
 
-  Soit $\lambda$ la mesure de Lebesgue sur l'intervalle $[-1,1]$, et $g=\1_{[0,1]}$.  On définit $f_n(x)$ par
+  Soit $\lambda$ la mesure de Lebesgue sur l'intervalle $[-1,1]$, et $g=\1_{[0,1]}$. On définit $f_n(x)$ par
   \[
     f_n(x) =
       \begin{cases}
@@ -204,42 +205,23 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
 \end{exo}
 
 % -----------------------------------------------
-\begin{exo}
+\begin{exo}\emph{(Contradiction avec Fatou ?)}
 
-  On note $\lambda$ la mesure de Lebesgue. Soit $f_n=-\frac1n\mathds{1}_[0,n]$, $n\in\N^{*}$ et $f=0$. Montrer que $\suite[n\in\N^{*}]{f_n}$ converge uniformément sur $\R$ vers $f$ mais que
+  On note $\lambda$ la mesure de Lebesgue. Soit $f_n=-\frac1n\1_{[0,n]}$, $n\in\N^{*}$ et $f=0$. Montrer que $\suite[n\in\N^{*}]{f_n}$ converge uniformément sur $\R$ vers $f$ mais que
   \[
     \liminf_{n\to +\infty} \int_{\R} f_nd\lambda <\int_{\R}fd\lambda.
   \]
-  Pourquoi est-ce que cela ne contredit pas le lemme de Fatou ?
-\end{exo}
-
-% -----------------------------------------------
-\begin{exo}
-
-  Soit $f:\R\to\R_+^*$ une fonction réelle, mesurable, strictement positive, et intégrable sur $\R$. Soit $a$ un réel strictement positif. Montrer qu'il existe une constant $c$, que l'on déterminera, telle que
-  \[
-    \lim_{n\to\infty}\int_\R n\log\pa{1+\pa{\frac{f(x)}{n}}^a}dx =
-    \begin{cases}
-      \infty & \text{ si }0<a<1, \\
-      c      & \text{ si }a=1,   \\
-      0      & \text{ si }1<a
-    \end{cases}.
-  \]
+  Est-ce que cela contredit le lemme de Fatou ?
 \end{exo}
 
-
-% ===============================================
-\section{Applications}
-% ===============================================
-
 % -----------------------------------------------
 \begin{exo}\emph{(Lemme de Fatou et quasi-domination)}
 
   Soit $\suiteN{f_n}$ une suite de fonctions mesurables convergeant simplement $\mu$-presque partout vers $f$. Soient $h$ et $\suiteN{g_n}$ des fonctions positives et $\mu$-intégrables. On suppose que $|f_n|\leq g_n+h$, et que $\lim_{n\to\infty}\int_E g_nd\mu= 0$
   \begin{enumerate}
     \item En utilisant le Lemme de Fatou, montrer que $f$ est $\mu$-intégrable.
-    \item Montrer que $\liminf g_n=0$ $\mu$-presque-partout en utilisant le Lemme de Fatou.
-    \item Grâce au Lemme de Fatou et aux fonctions $h\pm f+\liminf g_n$, montrer que
+    \item Montrer que $\liminf g_n=0$ $\mu$-presque partout en utilisant le Lemme de Fatou.
+    \item En appliquant le Lemme de Fatou aux fonctions $g_n + h \pm f$, montrer que
     \[
       \limsup_{n\to\infty} \int_E f_nd\mu\leq \int_E fd\mu\leq \liminf_{n\to\infty} \int_E f_nd\mu,
     \]
@@ -247,45 +229,4 @@ Dans toute la fiche, on supposera  sauf mention contraire que l'on se trouve sur
   \end{enumerate}
 \end{exo}
 
-% -----------------------------------------------
-\begin{exo}
-
-  Soit $f:\R\to\R$ une fonction dérivable sur $[a,b]$, dont la dérivée $f'$ est bornée sur $[a,b]$. Montrer que $f'$ est intégrable sur $[a,b]$, et que
-  \[
-    \int_{a}^b f'(x)dx=f(b)-f(a).
-  \]
-  \indication{On pourra considérer $g_n(x)=n\1_{[a,b-1/n[}(x)\pa{f(x+1/n)-f(x)}$.}
-\end{exo}
-
-% -----------------------------------------------
-\begin{exo}\emph{(Mesures à densité)}
-
-  Soit $h:E\to[0,+\infty]$ une fonction mesurable. On définit $\nu$ sur $\mathscr{T}$ par $\nu(A)=\int_{A}hd\mu\coloneqq\int_E\1_{A}hd\mu$.
-  \begin{enumerate}
-    \item Verifier que $\nu$ est une mesure sur $(E,\mathscr{T})$.
-    \item Démontrer que si $A\in \mathscr{T}$ satisfait $\mu(A)=0,$ alors $\nu(A)=0$.
-    \item Soit $f: (E,\mathscr{T})\to(\R,\mathcal{B}(\R))$ mesurable. Montrer que $f$ est $\nu$-intégrable si et seulement si $fh$ est $\mu$-intégrable et que dans ce cas
-    \[
-      \int_E fd\nu=\int_E fh d\mu.
-    \]
-  \end{enumerate}
-\end{exo}
-
-% -----------------------------------------------
-\begin{exo}\emph{(Un critère d'intégrabilité)}
-
-  Supposons que $\mu$ est une  mesure finie, et $f:E\to\R$ une fonction mesurable. Pour tout $n\in\N$, on pose
-  \[
-    A_n=\ensemble{x\in E}{n\leq\abs{f(x)}}
-    \quad \text{ et }\quad
-    B_n=\ensemble{x\in E}{n\leq \abs{f(x)}< n+1}.
-  \]
-  Montrer que les propositions suivantes sont équivalentes.
-  \begin{enumerate}
-    \item La fonction $f$ est intégrable.
-    \item La série $\sum_{n\geq 0} n\mu(B_n)$ est convergente.
-    \item La série $\sum_{n\geq 0} \mu(A_n)$ est convergente.
-  \end{enumerate}
-\end{exo}
-
 \end{document}
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