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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!

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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.

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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

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