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

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Measure and Density
Posted: Feb 20, 2013 6:40 PM
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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



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