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 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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 index 750bbcf35d68393c8e1e9d2e1ecd157bcb87b269..eca34c1362cc7e8559e00b1d67607e0e8883400e 100644 GIT binary patch delta 19845 zcmdn<hxz7T<_$Y#>W$6p?6`_cic%AExhihGjrQ)B*)F*6dALvB?)JnJO5Ob4aSm=O z%sV5>z6mIta$NkC!$67U)6shS(40xWn`YZyUVZljThq~mZPh2THpRu;-e^D1UOM5+ zay~x3b8HW53o3qm=zLe}KfhmmzrB2mQ|rW2U*=D8j94Vf$Hm%UI<fxV9~D+d2DW_u zQ=JI`lK=kd=i8k;9RI)g$Hag8b}#pq(r}&NSXIBnUcN7^VrAzZVF8{MhyNb7kN3)y zsdezO@Lc)ww#}3O@B405N+sUE|M&I3{D}*MefV@PI}3>IEn*jNY~0i-clPw+*~hCD ze|~IFwvMTtA$M~A=G^7lf&x=}>o1#nEu8x4%<{}-OI>#v)ZYFcq<Q_`<-}S=cjp#W zueJQ{t~M9VRJt<Wt?A9)75s;xlKn%`BQ~vFbq}h)XwETw|M-E$$xF#!D{r+}`hJhy zb^g!M%Px(TIqb7pwm7}c>i@&gZ8DSf$o%#{8CNbdRW96PaY0!B=WO}(KVQ!FKUejw z?=I87%;a>_NvKk8x&LLg*SSZZYRidF|NCoRsn{N~EwApU?X_wW<r8T;yu5j1ltq+t 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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. 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zxuU!|MoT(1Y~2LQonG5cnHJ5x<6!e!He61cfByG2{ndxUogcogemFb-;q9Cs?<T+B z-B}U;?5<nQ-nV;AZ`SWM7p~XZTdwkZ+IKVeNB3FZ$DaK6@UGvqnlJBEf_?ApKKUx> zeeFrTIlJ4}=-QljG>ZLU_50)*_V;?57T)_UFjM-2-J~~#@AEc2b^Vk2R`P-U9#KEJ U=`zlYa$H7+23)GDuKsRZ0GiWQ$^ZZW 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 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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. 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z6|dVXS+`rJPX23Wd*{vgd%Z{Ad*54ovhMX=pFe*e*0#*h&;F37{LJ{?@kw{z*G|#- z_p-Lhr@TsDeYMnHfA#7!zf(TN$8_vpZ1;2BKW6)d-jUPQtQqCF3{6eAR8?L5-M9de C#_Wjz 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 index e626b0584c0319098687b1ba3ffba9d0b06e670c..913da2ea58062d9f5ac33330b45e9ab455fda60f 100644 GIT binary patch delta 20205 zcmeyemgUL@mJK^)>WwYz?6`_cic%AExhig5jXv9-vQ=>Z^Ys}|O%|`%TQ)UzTjC67 z5r<QnEw&1b?Js4eG&xyz9GE{}tKRnZ`@QqtMRp{FFMjttefry9>*n8Wm*;=B;OBnJ z3X6&^t(F&eAKop$^G)-A?e9m=``J_H6n1+3ODhREvgOgih-K~fxc8>kt1$;Qb~o}J z(@y7Qsr>(J^7}6j^Z(n{9Q*(O*EIQ^i^BrlWe+`k`?Ekr)WoPV$ZSGs`}Xtkb{V<0 z{~NCrT$1{E+3x!Le)lhp)fX;5zMlVo9!HhxlZKXg%&vM8^$eoC5?Ws>V&2;N$WNK} zlK=R|-IddqPuXwWd$_mjoTS&`i<$MU$5<W)%X#kckpHZ1o0<IQMq6REqWkHVt(QC6 z7bnhBIO=rqru(8T%iNWPU${54pW#ja@iX$)kM)Je?3SL(6!^HM;@s>F%kRAN-u>bF 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z*Hd?9(_P_UzH3`kIQis#Eq`5|q-!B({B%p4ci)4Yb5lMu>+WpoHV>}XJHGV^=enj{ z+!>0xi&VGtgkA88@~z#{XLdpD;MTJ-_J1<>>@>Y+Ci(u(vEP|{!&g6e=u^H~L8y8j zUv_)xcFwqcCFR_4$<NEU?)}|Wy;in*x7@++^WN2E|9`>P;o<Rp!q(*YoCCj-_0QJN l{>$@+`H!q!3TSaQ^nkV0=@t@<g4_lcCI(!ps;>TSTmX(&?L+_o 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 \[ \PP{\liminf A_n}=1. \] -- GitLab