515 – Caratheodory’s characterization of measurability (Homework 3)

This set is due Friday, April 27.

The goal of these problems is to prove Carathéodory‘s theorem that “extracts” a measure from any outer measure. In particular, when applied to Lebesgue outer measure, this construction recovers Lebesgue measure.

Recall that an outer measure on a set $X$ is a function $\lambda^*:{\mathcal P}(X)\to[0,+\infty]$ such that:

$\lambda^*(\emptyset)=0$ .
$A\subseteq B\subseteq X$ implies $\lambda^*(A)\le\lambda^*(B)$ .
For any $A_1,A_2,\dots$ subsets of $X$ , we have $\lambda^*(\bigcup_n A_n)\le\sum_n\lambda^*(A_n)$ .

Given a set $X$ and an outer measure $\lambda^*$ on $X$ , let $\Sigma$ denote the collection of subsets $E$ of $X$ with the property that

$\lambda^*(A)=\lambda^*(A\cap E)+\lambda^*(A\setminus E)$

for all $A\subseteq X$ .

Prove that $\Sigma$ is a $\sigma$ -algebra on $X$ .

This requires some work. You may want to proceed by stages:

First, check that $\Sigma$ is precisely the collection of sets $E$ such that, for any $A\subseteq X$ , we have
$\lambda^*(A)\ge\lambda^*(A\cap E)+\lambda^*(A\setminus E)$ .
Check that $\emptyset \in\Sigma$ , and that $\Sigma$ is closed under complements.
Check that $\Sigma$ is closed under finite unions. Conclude that it is also closed under set theoretic differences: If $E,F\in\Sigma$ , then $E\setminus F\in \Sigma$ .
The crux of the matter, of course, is to verify that $\Sigma$ is closed under countable unions. Accordingly, suppose that $E_n\in\Sigma$ for all $n$ , and let $A\subseteq X$ .

Let $E=\bigcup_n E_n$ , and note that $E=\bigcup_n F_n$ , where $G_n=\bigcup_{m\le n}E_m$ , $F_1=G_1=E_1$ and, recursively, $F_n=G_n\setminus G_{n-1}=E_n\setminus G_{n-1}$ for $n>1$ . (Note also that $G_n,F_n\in \Sigma$ for all $n$ .)

Then, for $n>1$ , $\lambda^*(A\cap G_n)=\lambda^*(A\cap G_{n-1})+\lambda^*(A\cap F_n)$ , and for all $n$ ,

$\lambda^*(A\cap G_n)=\sum_{m\le n}\lambda^*(A\cap F_m).$

Conclude that $\lambda^*(A\cap E)\le\lim_{n\to\infty}\lambda^*(A\cap G_n)$ . (Why does this limit exist?)

Also, prove that $\lambda^*(A\setminus E)\le\lim_{n\to\infty}\lambda^*(A\setminus G_n)$ . (Again, why does this limit exist?)

Conclude from these inequalities and item 1 that $E\in\Sigma$ . This concludes the proof that $\Sigma$ is a $\sigma$ -algebra.

Now let $\lambda:\Sigma\to[0,+\infty]$ denote the restriction of $\lambda^*$ to $\Sigma$ .

Prove that $(X,\Sigma,\lambda)$ is a measure space.

In view of what we have proved already, note that this “reduces” to prove that, whenever $E_1,E_2,\dots$ are pairwise disjoint elements of $\Sigma$ , then

$\lambda(\bigcup_n E_n)\ge\sum_n\lambda(E_n)$ .

With notation as before, check first that $\lambda(G_n)=\sum_{m\le n}\lambda(E_n)$ for all $n$ , and conclude.

Prove that $\lambda$ is in fact a complete measure. Recall that this means that any subset of a set of $\lambda$ -measure 0 is measurable and also has measure 0. In fact, check that if $\lambda^*(A)=0$ , then $A\in\Sigma$ , and conclude from this.
Suppose that $Y\subseteq X$ . Show that the restriction of $\lambda^*$ to ${\mathcal P}(Y)$ is an outer measure on $Y$ . Denote by $\Sigma_X,$ resp. $\Sigma_Y$ the set $\Sigma$ defined above, for $\lambda^*,$ resp. $\lambda^*\upharpoonright{\mathcal P}(Y)$ . Show that if $E\in\Sigma_X$ , then $E\cap Y\in\Sigma_Y$ . Suppose that $Y$ is measurable (i.e., that $Y\in\Sigma_X$ ). Is $\Sigma_Y=\Sigma_X\cap{\mathcal P}(Y)$ ? If so, is this the only case where equality holds?
Prove that if $\lambda^*$ is $m^*$ , Lebesgue outer measure on ${\mathbb R}^n$ , then $\lambda$ is precisely $m$ , Lebesgue measure on ${\mathbb R}^n$ . (This may be a bit easier for $n=1$ than in general.)

This entry was posted on Thursday, April 12th, 2012 at 2:15 pm and is filed under 515: Analysis II. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

	ปั่นสล็อต on 414/514 The theorems of Rieman…
	Richard Kimberly Hec… on Woodin’s proof of the se…
	Is there a Borel-mea… on 414/514 Examples of Baire clas…
	Simple bijectiveness… on 580 -Some choiceless results…
	Schur’s theore… on 580 -III. Partition calcu…
	Uncountable linearly… on 403/503 – Homework …
	What is a non-constr… on On strong measure zero se…
	function from Nmathb… on Hindsight
	Cauchy Schwarz Inequ… on 275 -Positive polynomials
	What is a Cantor-sty… on 580 -Some choiceless results…
	Question about Gener… on 580 -Some choiceless results…
	Set has Measure Zero… on On strong measure zero se…
	Non-measurable subse… on 580 -Cardinal arithmetic …
	hmeng1 on 414/514 The theorems of Rieman…
	580 -Cardinal arithm… on 580 -Cardinal arithmetic …

M	T	W	T	F	S	S
						1
2	3	4	5	6	7	8
9	10	11	12	13	14	15
16	17	18	19	20	21	22
23	24	25	26	27	28	29
30

A kind of library