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.