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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 19, 2013 3:37 PM
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On Feb 19, 1:39 pm, W^3 <82nd...@comcast.net> wrote:
> In article <Pine.NEB.4.64.1302161828050.5...@panix2.panix.com>,
>  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?

> > I have thought of fat Cantor sets, but I cannot see well how to
> > do it. Any suggestions, please?

> More interesting is to require 0 < m(S /\ I) < m(I) for all nonempty
> open intervals I contained in (0,1).

Let I_0, I_1, I_2, . . . be an enumeration of the open intervals with
rational endpoints. Construct a sequence of pairwise disjoint sets
F_0, F_1, F_2, . . . so that F_n is a nowhere dense set of positive
measure contained in the interval I_{floor(n/2)}. Let S be the union
of the sets F_n where n is even.

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