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.