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

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Rotwang

Posts: 1,679
From: Swansea
Registered: 7/26/06
Re: Measure and Density
Posted: Feb 16, 2013 10:16 PM
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On 17/02/2013 02:33, William Elliot wrote:
> Topology Q+A Board Ask An Analyst
>
> How can we find a measurable dense subset S of [0,1], with m(S) < 1,
> and such that for any (a,b) in [0,1], we have m(S /\ (a,b)) > 0?
>
> I have thought of fat Cantor sets, but I cannot see well how to
> do it. Any suggestions, please?


Let (q_n | n in N) be an enumeration of the rationals in [0, 1], and
consider the union of the sets (q_n - 1/2^{n + 3}, q_n + 1/2^{n + 3}) n
[0, 1] for n in N. I think that works.


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