```Date: Feb 22, 2013 12:53 AM
Author: quasi
Subject: Re: Measure and Density

W^3 wrote:>quasi wrote:>> Butch Malahide wrote:>> >W^3 wrote:>> >>>> >> Is it possible that there exist 0 < c < d < 1 such that >> >> cm(I) < m(S /\ I) < dm(I) for all nonempty open intervals >> >> I contained in (0,1)?>> >>> >No. If S is a (Lebesgue) measurable subset of the real line >> >with m(S) > 0, and if d < 1, then there is a nonempty interval >> >I such that m(S /\ I) > dm(I). Sometime in the previous >> >millennium I took a class in measure theory, using the >> >textbook by Halmos, and I recall that this was proved in an >> >early chapter.>> >>> >More is true:>> >>> >http://en.wikipedia.org/wiki/Lebesgue's_density_theorem>> >> A possibly related question ...>> >> Prove or disprove:>> >> If A,B are measurable subsets of [0,1] such that >> m(A /\ I) = m(B /\ I) for all open intervals I contained in>> [0,1], then m(A\B) = 0.>>True, it follows from: If E is a measurable subset of [0,1], >then for a.e. x in [0,1], I think you meant to say "for a.e x in E".>m(E /\ (x-d,x+d))/(2d) -> 1 as d -> 0. Apply this to A\B to >obtain a contradiction if m(A\B) > 0. Yes, I see it now. Thanks.>There may be a more elementary way to see it.quasi
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