Problem from Folland : based on Lebesgue Density Theorem: Let $D_{E}(x) = \lim_{r\to 0}\frac{\mu(E\cap B(r,x))}{\mu(B(r,x))}$ whenever it exists. Find examples of $E$ and $x$ such that $D_{E}(x)$ is a given number $\alpha \in (0,1)$ , or such that $D_{E}(x)$ does not exist. ($X = \mathbb{R}^n$,$\mu$ is Lebesgue measure)
Lebesgue Density Theorem
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2+5 for a problem copied from a textbook? Really? http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question – 2011-11-01
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0a problem is a problem!! doesnt matter if its textbook...!! – 2011-11-01
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0As Nate is probably implying or suggesting, you need to show more indications of where you are stuck, which parts of the question you do or don't understand, and so on – 2012-01-25
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0Why does this question has 6 negative votes? There are a lot of questions more stupid with a lot more positive votes... http://math.stackexchange.com/questions/54506/is-this-batman-equation-for-real – 2013-04-03
2 Answers
For the second part, let $B_n=B(1/n,0)=(-1/n,1/n)\subseteq\mathbb{R}$ for every $n\in\{1,2,\ldots\}$ and $$E=\bigcup_{n=1}^\infty(B_{(2n-1)!}\setminus B_{(2n)!}).$$
If $n$ is odd, then $B_{n!}\setminus B_{(n+1)!}\subseteq E$. Hence $$\frac{\mu(E\cap B_{n!})}{\mu(B_{n!})}\geq\frac{\mu(B_{n!}\setminus B_{(n+1)!})}{\mu(B_{n!})}=\frac{2/n!-2/(n+1)!}{2/n!}=1-\frac{1}{n}\longrightarrow 1$$ and we see that the Lebesgue upper density of $E$ at $0$ is $1$.
On the other hand, if $n$ is even, then $E\cap B_{n!}\subseteq B_{(n+1)!}$. Hence $$\frac{\mu(E\cap B_{n!})}{\mu(B_{n!})}\leq\frac{\mu(B_{(n+1)!})}{\mu(B_{n!})}=\frac{2/(n+1)!}{2/n!}=\frac{1}{n+1}\longrightarrow 0$$ and we see that the Lebesgue lower density of $E$ at $0$ is $0$.
Since the upper and lower density of $E$ at $0$ differ, the density of $E$ at $0$ does not exist.
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0Thanks for the nice solution! – 2013-01-11
For the first part, consider the region inside an angle centred in $x$ (with infinite semilines) with angle $\theta $ such that $\theta/2\pi=\alpha$. Geometrically it is obvious that the ratio is always equal to $\alpha$, and so the limit is $\alpha$.
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1For those interested in this sort of thing, an extreme example is given in the following paper: Allan deCamp, *The construction of a Lebesgue measurable set with every density*, Real Analysis Exchange 16 (1990-91), 344-348. – 2011-10-31
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0For a one dimensional example can I just project the circle to $\mathbb{R}$ like $[-\alpha n, \alpha n]$? – 2013-04-01