151 lines
11 KiB
Markdown
151 lines
11 KiB
Markdown
---
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excalidraw-plugin: parsed
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tags:
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- excalidraw
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excalidraw-open-md: true
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---
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up:: [[mesure positive d'une application|mesure]]
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#maths/intégration
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> [!definition] [[mesure image]]
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> Soit $(E, \mathcal{A}, \mu)$ un [[espace mesuré]]
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> Soit $(F, \mathcal{B})$ un [[espace mesurable]]
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> Soit $f: E \to F$ mesurable de $(E, \mathcal{A})$ dans $(F, \mathcal{B})$
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> Notons $\nu$ l'application de $\mathcal{B}$ dans $\overline{\mathbb{R}}_{+}$ définie :
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> $\begin{align} \nu: \mathcal{B} &\to \overline{\mathbb{R}}_{+}\\ B &\mapsto \mu(f^{-1}(B)) \end{align}$
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> $\nu$ est une [[mesure positive d'une application|mesure]] appelée **mesure image de $\mu$ par $f$**
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^definition
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`$= "![[" + dv.current().file.name + ".svg|700]]" `
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# Propriétés
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> [!proposition] La mesure image est bien une mesure
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> Soit $(E, \mathcal{A}, \mu)$ un [[espace mesuré]]
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> Soit $(F, \mathcal{B})$ un [[espace mesurable]]
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> Soit $f: E \to F$ mesurable de $(E, \mathcal{A})$ dans $(F, \mathcal{B})$
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> $\begin{align} \nu: \mathcal{B} &\to \overline{\mathbb{R}}_{+}\\ B &\mapsto \mu(f^{-1}(B)) \end{align}$
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> $\nu$ est une mesure de $(F, \mathcal{B})$
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> > [!démonstration]- Démonstration
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> > 1. Soit $(B_{n})_{n\in\mathbb{N}}$ une suite d'éléments 2 à 2 disjoints de $\mathcal{B}$
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> > $\displaystyle\nu\left( \bigcup _{n=0}^{+\infty} B_{n} \right) = \mu\left( f^{-1}\left( \bigcup _{n=0}^{+\infty}B_{n} \right) \right) = \mu\left( \bigcup _{n=0}^{+\infty} \underbrace{f^{-1}(B_{n})}_{A_{n}} \right)$
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> > $\forall n \in \mathbb{N}, \quad A_{n} \in \mathcal{A}$
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> > Si $n \neq p$ alors $f^{-1}(B_{n}) \cap f^{-1}(B_{p}) = \emptyset$
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> > Si $x \in f^{-1}(B_{n}) \cap f^{-1}(B_{p})$, alors $f(x) \in B_{n}$ et $f(x) \in B_{p}$
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> > or, $B_{n} \cap B_{p} = \emptyset$
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> > ainsi, $\displaystyle\nu\left( \bigcup _{n=0}^{+\infty}B_{n} \right) = \sum\limits_{n=0}^{+\infty} \mu(f^{-1}(B_{n})) = \sum\limits_{n=0}^{+\infty}\nu(B_{n})$
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> > Et donc $\nu$ est bien une mesure sur $\mathcal{B}$
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# Exemples
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> [!example] Exemple 1
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> Sur l'[[espace mesuré]] $([0; 1], \mathcal{B}([0; 1]), \lambda _{[0; 1]})$
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> Soit la tribu $\mathcal{B} = \{ \emptyset, \{ 0 \}, \{ 1 \}, \{ 0, 1 \} \}$
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> Soit $0 \leq a < b \leq 1$ on a $\lambda _{[0; 1]}(]\![ a, b ]\![) = b - a$
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> Soit $p \in [0; 1]$
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> Soit l'application :
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> $\begin{align} \mathbb{1}_{[0, p]} : [0; 1] &\to \{ 0; 1 \}\\ x &\mapsto \begin{cases} 0 \text{ si } x \in [0; p]\\ 1 \text{ sinon} \end{cases} \end{align}$
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> ([[fonction indicatrice]] de $[0; p]$)
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> On cherche la mesure image de $\lambda _{[0; p]}$ par $\mathbb{1}_{[0; p]}$
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> C'est une mesure sur $\mathcal{B}$. Notons la $\nu$.
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> $\nu(\{ 1 \}) = \lambda _{[0; 1]}((\mathbb{1}_{[0; p]})^{-1}(\{ 1 \})) = \lambda _{[0; p]}([0; p]) = p$
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> Pour tout $x \in [0; 1]$
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> $\mathbb{1}_{[0; p]} = 1 \iff x \in [0; p]$, donc :
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> $\nu(\{ 0 \}) = \lambda _{[0; 1]}((\mathbb{1}_{[0; 1]})^{-1}(\{ 0 \})) = \lambda _{[0; 1]}(]p; 1]) = 1-p$
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> $\nu(\{ 0; 1 \}) = \nu(\{ 0 \} \cup \{ 1 \}) = p+1 - p = 1$
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> Donc, $\nu$ est la [[mesure de Bernoulli]] de paramètre $p$ :
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> $\nu = p\delta_1+(1-p)\delta_0$
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>
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> [!example] Exemple 2
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> Soit $\nu = \frac{1}{6}d_0 + \frac{1}{3}d_1 + \frac{1}{2}d_2$
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> Soit $f: [0; 1] \to \{ 0, 1, 2 \}$ telle que $\nu$ soit l'image de $\lambda _{[0; 1]}$ par $f$
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> ![[mesure image 2024-09-18 15.26.58.excalidraw|500]]
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%%
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# Excalidraw Data
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## Text Elements
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## Embedded Files
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adb2e6d5f0fd60f27aefc6aea6ed09461088e5ee: $$E$$
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e96807732150d61ddfa7c8a0c877a47cbc3b3236: $$F$$
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ee89b59b1a1e5a14b6c025939a698bbac5f376a9: $$f^{-1}(B)$$
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e30129f05a01a4a4a7b432e0291c53bf6dcbb40e: $$B
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$$
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e2b8d6503e865684f3d10c742442f5b01f2337d6: $$\mu(B)$$
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5da2f1c2ab6d8be668e628b41596152fe3d036e7: $$\nu(B) = \mu(f^{-1}(B))$$
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## Drawing
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```compressed-json
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RzFkDX+xHSJH78NgHMNwJvxa9PXt2gPv/6EIFACB8f6QJH3H3YF1ggNgDkL0NPXABX1X9PZoLX25xHA/4QIwGwiH/gDD4dN7sYQYID/2HDb8/oUAAwCbwGTc0qW5rTirSAsw/8/+AAy0p2nADDJjSYMKsGNkbBAA
|
|
```
|
|
%% |