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The other day I was browsing the site and found the question. I was trying to follow up with Topologieeeee, but clearly [s]he has not shown up for quite a while. So I wonder if anybody knows where to find the proof of the FACT referred in [s]he's question?

Thanks.

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The FACT, from the old post, is the following:

Let $E$ be a Lebesgue measurable subset of $\mathbb{R}^n$. Almost every $x\in E$ satisfies $\lim\limits_{m(B)\to 0,~x\in B}\frac{m(B\cap E)}{m(B)}=1$ i.e. limit is taken over the ball $B$ containing $x$ with shrinking it.

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    Could you please add "*the FACT*" to your question to make it self-contained? This is called the [Lebesgue density theorem](http://en.wikipedia.org/wiki/Lebesgue's_density_theorem) and can be found e.g. in Rudin's *Real and Complex Analysis*. If you can access it, you can also have look at [this recent proof](http://www.jstor.org/stable/2695333) by C.-A. Faure.2011-05-14
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    This is exercise 25 on page 100 of Folland, Real Analysis, chapter "Differentiation on Euclidean Space".2011-05-14

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This is called the Lebesgue Density Theorem. With that knowledge in hand it should be easy to search for and find a proof. I have made a nice proof, due to C.-A. Faure, from a recent Monthly article available here.

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    I'm very glad to see you're back! This is great.2011-05-14