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

 Messages: [ Previous | Next ]
 W^3 Posts: 29 Registered: 4/19/11
Re: Measure and Density
Posted: Feb 20, 2013 5:46 PM

In article
Butch Malahide <fred.galvin@gmail.com> wrote:

> 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.

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)?

Date Subject Author
2/16/13 William Elliot
2/16/13 Rotwang
2/16/13 quasi
2/16/13 quasi
2/16/13 Butch Malahide
2/19/13 W^3
2/19/13 Butch Malahide
2/20/13 W^3
2/20/13 Butch Malahide
2/21/13 David C. Ullrich
2/21/13 quasi
2/21/13 W^3
2/22/13 quasi
2/22/13 Butch Malahide
2/23/13 David C. Ullrich