Date: Feb 21, 2013 11:09 PM
Author: quasi
Subject: Re: Measure and Density
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

quasi