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<math>\forall ~ x,y \in [0,1] (x \leq y)</math>,
 
<math>\forall ~ x,y \in [0,1] (x \leq y)</math>,
  
<math> |F(y)-F(x)|=\int_{x}^{y}f(t)dt=\int_{0}^{1}f(t) \chi_{[x,y]}(t) dt~ \stackrel{\rm Holder} {\leq} ~\left(\int_{0}^{1}|f(t)|dt\right)~||\chi_{[x,y]}||_{\infty} = M|x-y| </math>
+
<math> |F(y)-F(x)|=\int_{x}^{y}f(t)dt=\int_{0}^{1}f(t) \chi_{[x,y]}(t) dt~ \stackrel{\rm Holder} {\leq} ~\left(\int_{0}^{1}|f(t)|dt\right)~||\chi_{[x,y]}||_{\infty} = M|x-y| </math>.
  
 
Hence <math>~F</math> is a Lipschitz map, which preserves measurability. This proves (a).
 
Hence <math>~F</math> is a Lipschitz map, which preserves measurability. This proves (a).

Revision as of 18:37, 21 July 2008

9.9. Let $ f \in L^{1}([0,1]) $ and let $ F(x)=\int_{0}^{x}f(t)dt $. If $ ~E $ is a measurable subset of $ ~[0,1] $, show that

(a) $ F(E)=\{y: \exist ~x \in E $ with $ ~y=F(x)\} $ is measurable.

Proof.

Let $ \int_{0}^{1}|f(t)|dt=M<\infty $.

$ \forall ~ x,y \in [0,1] (x \leq y) $,

$ |F(y)-F(x)|=\int_{x}^{y}f(t)dt=\int_{0}^{1}f(t) \chi_{[x,y]}(t) dt~ \stackrel{\rm Holder} {\leq} ~\left(\int_{0}^{1}|f(t)|dt\right)~||\chi_{[x,y]}||_{\infty} = M|x-y| $.

Hence $ ~F $ is a Lipschitz map, which preserves measurability. This proves (a).


(b) $ m(F(E)) \leq \int_{E}|f(t)| dt $.

Proof.

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