3
$\begingroup$

I noticed the following exercise in a measure theory text. It strikes me as interesting and I wanted to see if any visitors to the site today could assist me in thinking through it.

Suppose $S \subseteq \mathbb{R}$ is a Lebesgue measurable set of positive measure $\mu(S) > 0$. Then for each $\varepsilon \in (0, \mu(S))$, there exists $C \subseteq S$ such that $C$ is compact and $\mu(C) = \varepsilon$.

I am aware of a result saying that the measure of a Lebesgue measurable set $S$ can be approximated by the measure of some open set containing $S$ and that of some compact set contained in $S$, and I was trying to apply that here for about an hour, but to no avail. I would really appreciate some help.

  • 0
    @Shawn: I see why $f$ is increasing, but I am having trouble with continuity. Is there a particular formulation of continuity that would be best for seeing that $f$ is continuous? Also, is it standard in measure theory (at least with the Lebesgue measure) that you can define continuous functions in terms of the measure of a particular set, and if so, are you aware of standard texts where this is discussed (I ask since I have seen at least one other post on the site where defining such a continuous function solved a visitor's problem)?2012-03-02

1 Answers 1

2

Since Lebesgue measure is inner regular, you can find a compact set $K \subset S$ such that $\mu(K)>\epsilon$. Since $K$ is compact, it is in particular bounded and we can find an interval $[a,b]$ such that $K \subset [a,b]$.

Now define $f:[a,b]\rightarrow \mathbb{R}$ by $f(x)=\mu([a,x] \cap K)$. It is clear that $f$ is increasing. Notice that (if y>x) we have

$f(y)-f(x)=\mu((x,y] \cap K) \leq \mu((x,y])=y-x,$ from which it immediately follows that $f$ is continuous.

Since $f$ is continuous and $f(a)=0$ and $f(b)=\mu(K)>\epsilon$, it follows by the intermediate value theorem that there is an $x_0$ such that $f(x_0)=\epsilon$. The set $C=[a,x_0]\cap K$ will be a compact set in $S$ with measure $\epsilon$.

I'm not sure if it is a "standard technique" to do something like this. I remember when I took measure theory this was an occasionally useful one to solve homework problems, but I don't remember it being discussed in the text I used (Rudin's text).

  • 0
    Tha$n$ks for the clar$i$ficatio$n$ Shawn, $i$t really helped!2012-03-02