Posted by J. Holmer on December 2, 2007, 6:14 pm, in reply to "Re: Extension?"
Board Administrator
Proof: Since 0 is a point of Lebesgue density of E, we know that
\lim_{B containing 0, |B|-->0} m(E\cap B)/m(B) = 1.
Take the sequence of balls B(0,1/n). From the above, for n sufficiently large, we know that m(E\cap B(0,1/n)) \geq 1/2 * m(B(0,1/n)) >0. In particular, E\cap B(0,1/n) is nonempty. Take x_n\in E\cap B(0,1/n).
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