
Re: Measure and Density
Posted:
Feb 20, 2013 6:40 PM


On Feb 20, 4:46 pm, W^3 <82nd...@comcast.net> 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

