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