update
This commit is contained in:
@@ -25,7 +25,7 @@ up:: [[mesure positive d'une application|mesure]]
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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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> $\nu$ 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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@@ -64,6 +64,23 @@ up:: [[mesure positive d'une application|mesure]]
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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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@@ -73,79 +90,84 @@ up:: [[mesure positive d'une application|mesure]]
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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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834f4f03db2630a85af1da27414acaf4a96e2d6b: $$(E, \mathcal{A}, \mu)$$
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e96807732150d61ddfa7c8a0c877a47cbc3b3236: $$F$$
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2a25a46d8b55d25eb04e97f32205978c7cfc4a56: $$(F, \mathcal{B})$$
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ee89b59b1a1e5a14b6c025939a698bbac5f376a9: $$f^{-1}(B)$$
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93dd76f01859baaa41c85298a66215822c042d56: $$f$$
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e30129f05a01a4a4a7b432e0291c53bf6dcbb40e: $$B
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$$
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6ebd9d18c3e73ee6fa41fa7fe0561392117d6789: $$A = f^{-1}(B)$$
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e2b8d6503e865684f3d10c742442f5b01f2337d6: $$\mu(B)$$
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811c6101906f566736a7299004cd26f5fd73c338: $$B = f(A)$$
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5da2f1c2ab6d8be668e628b41596152fe3d036e7: $$\nu(B) = \mu(f^{-1}(B))$$
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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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||||
```
|
||||
%%
|
||||
Reference in New Issue
Block a user