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