10
$\begingroup$

Every set mentioned is a subset of the real numbers.

Let $m^*(C)$ denote the outer measure of a set $C$. Let $E$ be $any$ set and $A,B$ be measurable, disjoint sets. I'm trying to show that $$m^*(E\cap (A\cup B))=m^*(E\cap A)+m^*(E\cap B).$$

Proof: ($\le$) follows by the countable subadditivity of the outer measure since $$E\cap (A\cup B)= (E\cap A)\cup (E\cap B).$$

Here's where I get stuck:

($\ge$) My attempts have reduced to something of the form:

  1. There are bounded open sets $G_1, G_2$ containing $E\cap A, E\cap B$, respectively, such that $$m(G_1)\ge m^*(E\cap A),\quad m(G_2)\ge m^*(E\cap B).$$ Hence $$m^*(E\cap A)+m^*(E\cap B)\le m(G_1)+m(G_2).$$ And I would like to extend this inequality to $m(G_1\cup G_2)$ but I know that's not even true, especially since the sets $G_1, G_2$ may not even be disjoint.
  2. I also tried de la Vallée-Poussin Criterion: Let $\epsilon>0$. Since $A, B$ are measurable, there are closed subsets $F_1, F_2$ of $A,B$ respectively, such that $m^*(A\cap E - F_1)+ m^*(B\cap E-F_2)<\epsilon$. Even if I could show, $$|m^*(E\cap (A\cup B)-[(F_1\cup F_2))+ m^*(A\cap E - F_1)+ m^*(B\cap E-F_2)]|<\epsilon.$$

    I'm not sure what that would mean.

What I know:

  • Measure has only been defined for bounded sets.

  • A bounded set $A$ is $measurable$ if its outer and inner measures are equal; if so, the measure of $A$ is the common value of these measures.

  • Differences, countable unions, countable intersections of measurable sets are measurable.

  • The union of a set of pairwise disjoint measurable sets is measurable, with the measure of the union equal to the sum of the measures of the sets in the union.

  • Outer and inner measures are monotone increasing functions.

  • Countable subadditivity for outer measure, which states that if $A$ is a countable or finite union of sets $A_i$ then $m^*(A)\le \sum m^*(A_i)$.

  • De la Vallée-Poussin Criterion, which states that a bounded set $A$ is measurable iff for every $\epsilon >0$ there is a closed set $B\subset A$ such that $m^*(A-B)< \epsilon$.

  • For any bounded set $B$, I can always find a set $C$ that is a countable intersection of open sets for which $B \subset C$ and $m^*(B)=m^*(C)$.

  • If $A$ and $B$ are measurable sets, then $m(A\cup B) + m(A\cap B) = m(A) + m(B)$.

  • If $A$ is bounded and $I$ is an open interval containing $E$, then $m^*(E) + m_*(I-E) = m(I)$.

  • 0
    Why are subsets of measurable sets bounded? Consider for example $]0,\infty[\subset \mathbb{R}$, where $\mathbb{R}$ is measurable and $]0,\infty[$ is unbounded.2012-08-13
  • 0
    So far my notes have only defined measurability for bounded sets. Haven't gotten there yet, but thanks for the insight.2012-08-13
  • 0
    Perhaps [useful](http://math.stackexchange.com/a/81875/8271)2012-08-17
  • 1
    Notice that Caratheodory Criterion is equivalent to the problem you pose. You can get a proof of the Caratheodory Criterion for example from the Wheeden & Zygmund's _Measure and Integral_ book, to get some inspiration.2012-08-17
  • 0
    @leo Thanks for the link you sent, but the equation in that question has both inner and outer measure. Even though my set $A$ is measurable, we cannot conclude that $A\cap E$ is. I proved the Caratheodory Criterion by using the result I am trying to prove. My goal is to create/find a solution to the reverse inequality that I posed, rather than using the proof of Caratheodory (which is a later problem in my notes).2012-08-17
  • 0
    My last resort is to note that, as you said, these two problems are equivalent and then to give a proof of Caratheodory. Since Caratheodory comes later in my notes, I am assuming this problem can be done without Caratheodory's result.2012-08-17
  • 0
    That's the proof I have in mind. They are equivalent so you can prove your assertion by proving Cratheodory "from scratch". I suggested that way in order to get some inspiration to proceed the way you want. Let me write something. Perhaps I come back with an answer in while.2012-08-18
  • 0
    I like Byron's edit.2012-08-18
  • 0
    Byron, nicely presented. Thanks.2012-08-19

3 Answers 3

0

It can be shown that there exists nonvoid disjoint subsets A,B of R such that m*(A U B) is strictly less than m*(A) + m*(B) using the Axiom of Choice.

  • 0
    You still need to elaborate2016-05-06
3

Put $F=E\cap (A\cup B)$ then, since $A$ is $m^*$-measurable, we have that $$ m^*(F)= m^*(F\cap A) + m^*(F\cap A^c) = m^*(E\cap A)+m^*(E\cap B) $$ since $A\cap B =\emptyset$

  • 0
    I want to use the equation in my problem to prove the Caratheodory Crtieria / the equation you just used, so I don't want to assume this equation is true yet.2012-08-14
  • 1
    @Broseph: I suggest making explicit in the question your definition of measurability, as well as the results that are available so that a more appropiate answer may be given.2012-08-14
2

We start with a little Lemma:

Lemma. Let $E\subseteq \Bbb R$. If $H\supseteq E$ is a $G_\delta$ set (countable intersection of open sets) such that $$m(H)=m^\ast(E),$$ then for every $C\subseteq\Bbb R$ $$m^\ast(H\cap C)=m^\ast(E\cap C).$$

Proof. Let $C\subseteq\Bbb R$. In the following the superscript $^c$ means complement. $$\begin{align*} m^\ast(H\cap C) &\leq m^\ast(H\cap C\cap E\cap C)+m^\ast((H\cap C)\setminus (E\cap C))\\ &= m^\ast(E\cap C) + m^\ast((H\cap C)\cap (E\cap C)^c)\\ &= m^\ast(E\cap C) + m^\ast(C\cap (H\setminus E))\\ &\leq m^\ast(E\cap C) + m^\ast(H\setminus E)\\ &= m^\ast(E\cap C) \end{align*}$$ The inequality $m^\ast(H\cap C)\geq m^\ast(E\cap C)$ comes free by the monotony of the outer measure since $H\supseteq E$.

Proof of $m^\ast(E\cap (A\cup B))\geq m^\ast(E\cap A)+m^\ast(E\cap B)$.

Pick $H\supseteq E$ a $G_\delta$ set so that $m(H)=m^\ast(E)$. Then $$\begin{align*} m^\ast(E\cap (A\cup B)) &= m^\ast(H\cap (A\cup B)) &&\text{by the Lemma}\\ &= m(H\cap A) + m(H\cap B) &&\text{($^\ast$)}\\ &\geq m^\ast(E\cap A) + m^\ast(E\cap B) &&\text{by the monotony of the outer measure.} \end{align*}$$ ($^\ast$) because here we are dealing with measurable sets (of finite measure).

Observation. Notice that such a $G_\delta$ set $H$ always exist even if $E$ is unbounded.

  • 0
    I can see that such a set $H$ with the same outer measure of $E$ will exists. But how do you know that the countable intersection $H$ will be always be measurable?2012-08-19
  • 0
    If you can prove that open sets are measurable then that's a corolary. Do you can?2012-08-21
  • 0
    Yes, but the intersection of open sets isn't always open. Take $A_n = (-1/n,1/n)$ in the real line. Then $\cap A_n$ is a singleton, which won't be open in the real line.2012-08-21
  • 0
    You're right, it's a singleton, a singleton have outer measure zero so it is measurable. In the other hand, if you can proof that an open set is measurable by the third bullet of **What you know** it follows that a countable intersection of measurable sets is always measurable.2012-08-21
  • 0
    Thank you, and thanks to everyone who contributed.2012-08-21
  • 0
    @Broseph are you fully satisfied? It's all clear?2012-08-21
  • 0
    Yes, everything follows.2012-08-21