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z=he^pr|b+!{l_zVzub?@=LNg>*P08|AKItSvitt`%18Mt-=98s?}xmJo{+ZY{j*2v zf8{A{`d_ns(u*&5o!q2q%UP;#l}ktdIq`jKg8t8UJm<W(&z*HdX9J&cjL)v=;tGti Q+~!6m=3J_(uKsRZ0Q`wMMgRZ+ 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} -- GitLab