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Math Forum » Discussions » sci.math.* » sci.math.independent

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

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Butch Malahide

Posts: 885
Registered: 6/29/05
Re: Measure and Density
Posted: Feb 16, 2013 11:45 PM
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On Feb 16, 8:33 pm, William Elliot <ma...@panix.com> 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?

Any dense open set will have the property that its intersection with
each interval has positive measure. Therefore, it will suffice to
construct a dense open subset S of (0,1) with m(S) < 1. In fact, you
can easily make your dense open set S have measure less that an
arbitrary positive number epsilon. If you do that for an infinite
sequence of epsilons tending to zero, and take the intersection, you
get a dense G_{delta} set of measure zero. Haven't you seen that
construction?

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