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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.

Edit

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 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 by C.-A. Faure. –  t.b. May 14 '11 at 18:19
    
This is exercise 25 on page 100 of Folland, Real Analysis, chapter "Differentiation on Euclidean Space". –  Bruno Stonek May 14 '11 at 18:21
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up vote 10 down vote accepted

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. –  Bruno Stonek May 14 '11 at 18:27
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