Open Bounded subset with non-zero measure boundary

[Kreizhn]

Homework Statement

Let m be the Lebesgue measure on $\mathbb R^d$, and define the open sets $O_n = \{ x \in \mathbb R^d : d(x,E) < \frac1n \}$ where
$$d(A,B) = \inf\{ |x-y| : x \in A, y \in B \}$$

1) Find a closed and unbounded set E such that $\lim_{n\to\infty} m(O_n) \neq m(E)$.

2) Find an open and bounded set E such that $\lim_{n\to\infty} m(O_n) \neq m(E)$.

The Attempt at a Solution

This is technically the second part to question, which was to show that if E is compact then the limits do in fact hold. I think I might have the first part. In particular, I've just chosen a countable number of copies of the cantor set C in $\mathbb R$. This set has zero measure, but $m(O_n) = \infty$ for every $n \in \mathbb N$ and so the limits don't converge.

My real issue is trying to find an open bounded set. My first thoughts were to try the complement to the cantor set $[0,1]\setminus C$ but this doesn't seem to produce the desired results. In particular, I know that we must have that $m(E) < m(O_n)$ by monotonicity, but since E is open (and we'll assume non-empty) then m(E) > 0, so there won't be any 0 = infinity mumbo jumbo here. In particular, it seems to me that the set I want will be one such that m(E) is not the same as m(cl E) where cl E is the closure of E. That is, the boundary of E will have non-zero measure. I just can't think of such a set.

Any help would be appreciated.

 

[mighty2000]

Hello,

Thank you for your post. I think you are on the right track with your approach for the first part. The Cantor set is indeed a good example of a closed and unbounded set that satisfies the given conditions.

For the second part, I suggest considering the set E = (0,1) \cup (2,3) in \mathbb{R}. This set is open and bounded, and has a finite measure of 2. However, as n goes to infinity, the measure of O_n will also go to infinity, as the distance between E and any point outside of E will approach 0. Therefore, the limits do not converge.

I hope this helps. Let me know if you have any further questions.