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Topic: Measure and Density
Replies: 14   Last Post: Feb 23, 2013 11:26 AM

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quasi

Posts: 9,903
Registered: 7/15/05
Re: Measure and Density
Posted: Feb 21, 2013 11:09 PM
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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



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