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Topic: Half Measure
Replies: 14   Last Post: Mar 25, 2013 5:03 PM

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JT

Posts: 1,170
Registered: 4/7/12
Re: Half Measure and Correction
Posted: Mar 25, 2013 4:09 PM
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On 25 mar, 14:07, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> On Sun, 24 Mar 2013 13:55:12 -0700 (PDT), Butch Malahide
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> <fred.gal...@gmail.com> wrote:

> >On Mar 24, 1:49 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> >> On Sun, 24 Mar 2013 10:37:46 -0700 (PDT), Butch Malahide
> >> > <fred.gal...@gmail.com> wrote:
> >> >On Mar 24, 10:53 am, David C. Ullrich <ullr...@math.okstate.edu>
> >> >wrote:

> >> >> [. . .]
> >> >> So the more interesting version of the question,
> >> >> in any case less trivial, amounts to this: Is there
> >> >> a measurable set D such that

>
> >> >> 0 < m(D intersect I) < m(I)
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> >> >> for every open interval I,
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> >> >Didn't we just have that thread?
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> >> Yes.
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> >> >http://groups.google.com/group/sci.math/msg/0cfe35786f2279f0?hl=en
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> >> >> and such that m(D intersect [0,1]) = 1/2 ?
>
> >> >OK, that's different.
>
> >> Precisely! heh.
>
> >But not so very different, is it?
>
> I was joking. Of course once we know that there exists
> D with 0 < m(D intersect I) < m(I) for every interval I it
> follows that there exists such a D with the second condition.
>
> By any of at least three arguments:
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> 1. The one I had in mind.
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> 2. The one you give below.
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> 3. By saying "Fine, now how in the world could
> it be that the value 1/2 is somehow excluded?"
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> >Let D be a measurable set such that
> >0 < m(D intersect I) < m(I) for every interval I. It will suffice to
> >find an interval I such that m(D intersect I)/m(I) = 1/2. Since m(D) >
> >0, there is an interval J such that m(D intersect J)/m(J) > 1/2;
> >likewise, since m{R\D) > 0, there is an interval H such that m(D
> >intersect H)/m(H) < 1/2. Since the function f(a,b) = m(D intersect
> >(a,b))/(b-a) is continuous on the connected domain {(a,b): a < b},
> >there is an interval I = (a,b) such that f(a,b) = 1/2.

I am not sure what you are onto here, but noone have forbidden
nullsamples, nor nullmembers of set for statistical use, not even
finitists.



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