Difference between revisions of "Jets9.9 OldKiwi" - Rhea

Revision as of 21:35, 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.

(Step 1) $ ~E=(a,b) $

First assume that $ f \geq 0 $. Since $ ~F $ is monotone and continuous, $ m(F(E))=F(b)-F(a)=\int_{a}^{b}f(t) dt = \int_{E}|f(t)| dt $.

In general, let $ ~ f=f^{+}-f^{-} $ and let $ F^{+}=(x)\int_{0}^{x}f^{+}(t)dt, ~F^{-}=\int_{0}^{x}f^{-}(t)dt $. Then, $ F(x)=F^{+}(x)-F^{-}(x) $.

$ m(F(E)) = \max_{x \in E}F(x)- \min_{x \in E}F(x) \leq \left( \max_{x \in E}F^{+}(x) - \min_{x \in E}F^{+}(t) \right)-\left( \max_{x \in E} F^{-}(x)-\min_{x \in E}F^{-}(x) \right) $

\int_{a}^{b}f^{+}(t) dt + \int_{a}^{b} f^{-}(t)dt =\int_{a}^{b}|f(t)|dt =\int_{E}|f(t)|dt </math>.

(Step 2) $ ~E $ : open set

Let $ E=\bigcup_{n=1}^{\infty}I_{n} $ when $ ~I_{n} $'s are disjoint open intervals, so that

$ m(F(E))=m(F(\bigcup_{n=1}^{\infty}I_{n}))=m(\bigcup_{n=1}^{\infty}F(I_{n})) \leq \sum_{n=1}^{\infty} m(F(I_{n})) \stackrel{(Step 1)}{\leq} \sum_{n=1}^{\infty} \int_{I_{n}}|f(t)| dt= \int_{E}|f(t)|dt $

(Step 3) $ ~E $ is a $ ~G_{\delta}-set $

We can write $ E= \bigcap_{n=1}^{\infty} G_{i} $ when $ ~G_{n} $'s are nested open sets($ ~G_{1} \supseteq G_{2} \supseteq ... $). Then,

$ m(F(E))=m(F(\bigcap_{n=1}^{\infty} G_{i})) \leq m(\bigcap_{n=1}^{\infty}F(G_{n}))=\lim_{n \to \infty}m(\bigcap_{i=1}^{n}F(G_{i}))=\lim_{n \to \infty}m(F(G_{n})) $

$ \stackrel{(Step 2)}{\leq} \lim_{n \to \infty} \int_{G_{n}}|f(t)|dt=\lim_{n \to \infty} \int_{0}^{1} |f(t)|\chi_{G_{n}}(t) dt \stackrel{\rm MCT} {=} \int_{0}^{1}|f(t)|\chi_{\bigcap_{n=1}^{\infty}G_{n}}(t)dt = \int_{0}^{1}|f(t)| \chi_{E}(t) dt = \int_{E}|f(t)| dt $

(Step 4) $ ~E $ is a measurable set.

From a characterization of measurability, $ ~E=H-Z $, where $ ~H $ is a $ ~G_{\delta} $ set and $ ~m(Z)=0 $. Then,

$ m(F(E))=m(F(H-Z)) \leq m(F(H)) \stackrel{(Step 3)}{\leq} \int_{H}|f(t)|dt = \int_{E}|f(t)|dt $

- JIN -