JT
Posts:
1,388
Registered:
4/7/12


Re: Half Measure and Correction
Posted:
Mar 25, 2013 4:09 PM


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 > > > > > > > > > > <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) > > >> >> for every open interval I, > > >> >Didn't we just have that thread? > > >> Yes. > > >> >http://groups.google.com/group/sci.math/msg/0cfe35786f2279f0?hl=en > > >> >> 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: > > 1. The one I had in mind. > > 2. The one you give below. > > 3. By saying "Fine, now how in the world could > it be that the value 1/2 is somehow excluded?" > > > > > > > > >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))/(ba) 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.

