2

In a question paper I got the following three questions.

Let $E \subset \mathbb{R}$ with $0 < \mu(E) < \infty$.

  1. Show that the function $f : \mathbb{R} \rightarrow \mathbb{R}$ with $f(x) = \mu(E \cap (-\infty, x])$ is continuous.
  2. Show that there exist a measurable set $F \subset E$ such that $\mu(F) = \frac{1}{3}\mu(E)$.
  3. Show that there exist a closed set $F \subset E$ such that $\mu(F) = \frac{1}{3}\mu(E)$.

Remark: we are working on the real line $(\mathbb{R})$, and $\mu$ is the Lebesgue measure ($\mathscr{M}$ is the set of measurable subsets of $\mathbb{R}$).

Edit:

Following the explanation of John Ma here's my attempt for question $(2)$.

Let $A_n = E \cap (-\infty, n]$ (increasing sequence of measurable sets)

and $B_n = E \cap (-\infty, -n]$ (decreasing sequence of measurable sets).

It follows that $$\mu(\bigcup_{n = 1}^{+\infty}A_n) = \lim_{n \to +\infty} \mu(A_n) = \lim_{n \to +\infty} f(n) = \mu(E)$$

and

$$\mu(\bigcap_{n = 1}^{+\infty}B_n) = \lim_{n \to +\infty} \mu(B_n) = \lim_{n \to +\infty} f(-n) = \mu(\emptyset) = 0.$$

Since $\lim_{n \to +\infty} f(n) = \mu(E)$, we choose $n_0$, $n_1$ (large enough) such that $$f(n_0) > \frac{9}{10} \mu{(E)} \;\; \text{and} \;\; f(-n_1) < \frac{1}{10} \mu{(E)}.$$

From the Intermediate Value Theorem it follows that $f$ is taking every value between $\frac{1}{10} \mu{(E)}$ and $\frac{9}{10} \mu{(E)}$. Therefore

$$\exists x_0 \in (-n_1, n_0) \;\; \text{such that} \;\; f(x_0) = \mu(E \cap (-\infty, x_0]) = \frac{1}{3}\mu(E).$$

Setting $F = E \cap (-\infty, x_0]$ we conclude that $$F \subset E, F \in \mathscr{M} \;\; \text{and} \;\; \mu(F) = \frac{1}{3}\mu(E).$$

glpsx
  • 2,122
  • So what did you try? –  Mar 04 '16 at 23:19
  • I was not able to do much, unfortunately. Our Real analysis teacher warned us of the difficulty of this problem. Any hint would be appreciated. – glpsx Mar 04 '16 at 23:32
  • Difficulty depends on what you are allowed to use. It would have been helpful if you had showed your thought process so as to give you a better hint. Can you write $f$ as a definite integral and what do you know about the continuity of a definite integral? – Daniel Akech Thiong Mar 04 '16 at 23:39

2 Answers2

2

Both $(1)$ and $(2)$ are not that difficult. For one, note that if $y >x$, then

$$\begin{split} f(y) - f(x) &= \mu((-\infty , y] \cap E) - \mu((-\infty, x],\cap E) \\ &= \mu((x,y]\cap E) \\ &\le |y-x|. \end{split}$$

So $f$ is indeed Lipschitz continuous. For $(2)$, this follows from $(1)$ and the intermediate value theorem. (What is $\lim_{x\to -\infty} f(x)$ and $\lim_{x\to \infty} f(x)$?)

The last one can be done using that $\mu$ is inner regular. Thus there is a closed set $G\subset E$ so that $$\frac 12\mu(E) <\mu(G) <\mu(E)$$ Now use the same idea as in $(2)$ to find $F' \subset G$ so that $\mu(F') = \frac 13 \mu(E)$. This can be done as $\frac 13 \mu(E) < \mu(G)$. Note that $F'$ will be a closed set.

Community
  • 1
  • 1
    How is it that $G\subset E$ and $\mu (G)=2\mu (E)$ ? – Svetoslav Mar 05 '16 at 00:12
  • Haha~ @Svetoslav Yes, thanks. Corrected. –  Mar 05 '16 at 00:14
  • I've edit my post and provided a solution for $(2)$ following your reasoning. Could you tell me if my solution is correct and complete? Thank you for your time and explanation, much appreciated. – glpsx Mar 05 '16 at 12:13
  • 1
    @VonKar : That's correct. Note that for $B_n$, you used not only that it is decreasing, but also that it's measure is finite. –  Mar 05 '16 at 12:14
  • Yes, $B_n$ is finite since $\mu(B_1) \leq \mu(E) < +\infty$. I'm working on $(3)$ now. I'll edit my answer if I can come up with a complete solution. – glpsx Mar 05 '16 at 12:24
0

Hint:

  1. Given a sequence of increasing measurable sets $E_n$, then $\mu( \cup E_n) = \sup \mu(E_n)$.
  2. Given a sequence of decreasing measurable sets $E_n$, all contained in some set of finite measure, $\mu( \cap E_n) = \inf \mu(E_n)$.

(I am recalling the statement after some time away from measure theory, so you should check in your book to make sure that there is no mistake. But I think they are correct.)

These are both standard (tricky) exercises, or else facts you have possibly proven in your class. They are relevant to your questions. Do you see why? What does continuity mean?

Elle Najt
  • 20,740