Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am having problems with doing preparatory exercises on Lebesgue density points, so I decided to ask you for hints. Here I'm posting two of them.

  1. A Borel set $E\subset \mathbb{R}^2$ has the property that each point from $[0,1]^2$ is a density point for $E$. Show that the Lebesgue measure of $E$ is not smaller than 1.

  2. Determine whether there exists a set $E\subset \mathbb{R}$ such that the set of Lebesgue density points of $E$ is equal to $\mathbb{R}\setminus \{0\}$

Ad 1.
I am trying to show that $\mu(E)\ge 1-\epsilon$ for arbitrary $\epsilon\gt 0$, but I cannot find a way. All I know after all my attempts is that for fixed $x\in [0,1]^2, \epsilon\gt 0$ we have by definition of L.d.p. $\mu(E\cap B(x,1/n))\gt (1-\epsilon)\mu(B(x,1/n))$ for sufficiently large $n$. Also, I'm familiar with Lebesgue differentiation theorem and its corollary which states that almost every point of $E$ is L.d.p., so the inequality holds also for $\mu$–a.e. $x\in E$.

Ad 2.
I suppose it is impossible for such a set to exist. If I assume the opposite, I see that $\lim_{r\to0} \frac{\mu(E\cap (-r,r))}{\mu((-r,r))}$ does not exist or exists and is equal to $c\in [0,1)$. However, I do not see how to derive contradiction.

All your help is strongly appreciated!

share|improve this question

1 Answer 1

up vote 3 down vote accepted

(1) If measure of E is less than one then $X = [0, 1]^2 \backslash E$ has positive area. So some point of $X$ is a density one point of $X$ and it follows that this point cannot be density one point of $E$.

(2) If $0$ is not a density one point of $E \subset \mathbb{R}$, then for some interval $I$ containing $0$, $I \backslash (E \cup \{0\})$ has positive measure hence a density one point.

share|improve this answer
    
Thanks for your help! I've discovered my problem with doing such elementary exercises is that I somehow cannot recognize which start-up observation/statement will lead to conclusion. I hope I'll get it someday. –  Kuba Helsztyński Jan 13 '13 at 14:40

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.