173 lines
12 KiB
Markdown
173 lines
12 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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#s/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 sur $(F, \mathcal{B})$, que l'on appelle **mesure image de $\mu$ par $f$**
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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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> [!example] Exemple 3
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> $(E, \mathcal{A}, \mu) = ([0; 1], \mathcal{B}([0; 1]), \lambda _{[0; 1]})$
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> $\begin{align} f : ]0; 1[ &\to \mathbb{R}^{+} \\ x &\mapsto -\ln(x) \end{align}$
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> Alors, soit $\nu$ la mesure image de $\mu$ par $f$ :
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> $$\begin{align}
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> \nu(]-\infty, t]) &= \lambda _{[0; 1]} (\{ x \in ]0; 1] \mid -\ln(x) \leq t \}) \\
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> &= \lambda _{[0; 1]} (\{ x \in ]0; 1] : x \geq e^{ -t } \}) \\
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> &= \lambda _{[0; 1]} ([e^{ -t }; +\infty[ \cap ]0; 1]) \\
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> &= \lambda _{[0; 1]} ([e^{ -t }; +\infty[) \\
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> &= \begin{cases} 0 \text{ si } t < 0 \\ 1-e^{ -t } \text{ si } t \geq 0 \end{cases}
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> \end{align}$$
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> ![[mesure image 2024-11-20 15.35.39.excalidraw]]
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>
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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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834f4f03db2630a85af1da27414acaf4a96e2d6b: $$(E, \mathcal{A}, \mu)$$
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2a25a46d8b55d25eb04e97f32205978c7cfc4a56: $$(F, \mathcal{B})$$
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93dd76f01859baaa41c85298a66215822c042d56: $$f$$
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6ebd9d18c3e73ee6fa41fa7fe0561392117d6789: $$A = f^{-1}(B)$$
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811c6101906f566736a7299004cd26f5fd73c338: $$B = f(A)$$
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c788d57471a52a30e34e72a1c31c096d236baaca: $$\begin{align}\nu(B) &= \mu(f^{-1}(B)) \\&= \mu(A) \end{align}$$
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e28573ea7722cca49ea84581133c4a77451f8cea: $$\mu(A)$$
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## Drawing
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```compressed-json
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9BlBJhRwOZ+3/hPg7h/h8BnASY267hQRJg/Csn529OdaRkkzQ8n7dPimIBBaxXhbIr/7TPpXSgNkEH2pOo5Q5QLQlQW2VW5CGnuwWxfwzRzDUChnPOTavQfR4fDWXocGzbAvShgvcIMwswcw8wxswvI5qxawY9GxmxWwhrbX6m4E6wYuxHIB4uy7+HAOdwaVkt/8UujnvXE7VqMvlRDl8x49A25HgPEPYkPH9TEb0IhuAnF7UBNRCXAm+OmrlQC8
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Cb05TL1geAJOMQhApPsMZPWg4AjAeAKBY5iBvCHLvC2ASY4RMBlBlBHQUg1Pzv7un7eXkyEzeanfhWHvSmnvymXuCyV5qn12gHdRJD1W+xDLKJjRKIgf6rmiHslCY3bSY+OyvOdNfQEffOke0/72RygurdNQVacwIplQpyKJmKXbOreB/FxoydrQS8eoyf/FcwjQSqwqf2OfnX6fgOkuD8xfWFWfDnMtO+TnIPuf+qy1ajcuEP8vu3OdjZtT+B1g
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4A4AEQuUGtoBBpMg1gQRaZIQGBCAEAKBgz09s/YfTgL/L/9+c1+YY0jh9AEQso0+uye6DbSgb/GpRx7+T++yz/jWAu8+kAD/nfwyA3ATcnvBmkAJECf97+j/GdoSDqBYdoBIAh/m/QgHX9kBzWe/i8B94PFrcSA2/lgIyCicV2Jnd/pgL2z38bgnAKADcCOgwg+MiA4AUQP0DUDsgcIMMnpSYEUCv+GQDmFgCgB/QiAygCOCEgQCnB8G5AwgZQIy
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Br96CbBbZlIJgEZBQQrBLEuwRqgo51KBA5QfoAYIcwDCF0ffswGwCYhYQMkVqBNBVrdgpcEUW+PZ2HSmCXQ+ANuGgEbD35QePTTar2APIQAjAbAAwA1i6AEBte90FjoUB0EoCcBlKB4lSBZqkBXUiAkMCQA4FGAuBEQ5IcQARAIA5sMvJIaQBIB3A2AKwVQbgE0DBBQ2JCEgEnj9j0wXQXEUgMoEDAAAKK0P+F4AyMOh7QkbAAEp/QLwBAMoFXAN
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Q1gjQloW1WoD8d+QUwyYb0MUThClB70OAdiBIFKROAgjIdH+0yADCaIBQ4An7CyBlCKh3ALXlJ2wDVAThpAbXgsDCLb80ApwtkEICgCGlLh2vXUuFU0DqwEA2AHIHCDCJwAihJQsIuUMsrrAfhhARgBzACH4AghtKbSOkHBFgYFgCBKAAYAMEBxg20SArv9jtB/RwRkI6ESQR9jgBHoHqYIDuGNhHggAA===
|
|
```
|
|
%% |