outer measure of the Vitali non-measurable set
Pouya D. Tafti
Just out of curiosity, does anyone have any idea regarding the
(Lebesgue) outer measure of the Vitali non-measurable set?
Regards,
Pouya
Philo D
Do you mean a set V in [0,1] containing one element from each
equivalence class for the relation: "x-y is rational" ? Because of
the arbitrary choices involved, it is best to call it "a Vitali
non-measurable set", not "the Vitali non-measurable set". The
outer measure could be any number in (0,1] .
Robert Israel
Pouya D. Tafti <pouya....@gmail.com> wrote:
>Just out of curiosity, does anyone have any idea regarding the
>(Lebesgue) outer measure of the Vitali non-measurable set?
See the thread "Vitali nonmeasurable" from January 1997.
<http://groups-beta.google.com/group/sci.math/browse_frm/thread/a1f91aa3b8ae80d8>
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Pouya D. Tafti
have limited mathematical background.
The example I had in mind was actually a set constructed by taking one
representative from each class defined by the equivalence
x~y <=> x-y=n\alpha (modulo-1 summation/subtraction) with \alpha
irrational,
although I now understand that the construction you discussed above is
also valid.
If I have understood your comments correctly, the outer measure of such
a set can be any number in (0,1] __depending on how the representatives
are chosen__. Is this right?
Thanks again and happy new year,
--
Pouya D. Tafti, McMaster University, Hamilton, Canada
http://grads.ece.mcmaster.ca/~pouya
p dot d dot tafti at ieee dot org
Dave L. Renfro
[sci.math: Dec 31 2004 3:26:15:000PM]
http://mathforum.org/discuss/sci.math/m/667141/667194
Pouya D. Tafti wrote:
>> Just out of curiosity, does anyone have any idea regarding the
>> (Lebesgue) outer measure of the Vitali non-measurable set?
Robert Israel wrote:
> See the thread "Vitali nonmeasurable" from January 1997.
>
> <http://groups-beta.google.com/group/sci.math/browse_frm/thread/a1f91aa3b8ae80d8>
An interestingly pathological related result that doesn't
seem to be very well known, possibly because the paper in
question doesn't appear in the three volume collection of
Sierpinski's papers ("Oeuvres Choisies", 1974-1976), is
proved in [1]. Sierpinski proves that there exists a pairwise
disjoint collection of perfect sets in [0,1] x [0,1] such
that if we choose a point from each set in this collection,
then we will always wind up with a nonmeasurable set that
has outer Lebesgue (planar) measure 1.
[1] Waclaw Sierpinski, "Sur un problème concernant les
familles d'ensembles parfaits", Fundamenta Mathematicae
31 (1938), 1-3.
The problem that motivated this paper is whether, given
any pairwise disjoint collection of perfect sets, it is
always possible to choose a point from each of them so as
to wind up with a set of measure zero. I don't know the
origin of this problem, but I do know that Sierpinski had
previously solved it in the negative by assuming the
continuum hypothesis.
Incidentally, if the original poster is interested, a lot
of fairly advanced results about Vitali sets is given in
A. B. Kharazishvili, "Nonmeasurable Sets and Functions",
North-Holland Mathematics Studies #195, Elsevier, 2004.
Dave L. Renfro
Pouya D. Tafti
> Incidentally, if the original poster is interested, a lot
> of fairly advanced results about Vitali sets is given in
>
> A. B. Kharazishvili, "Nonmeasurable Sets and Functions",
> North-Holland Mathematics Studies #195, Elsevier, 2004.
Thanks a lot. I requested the book through inter-library loan.
BTW if I am not mistaken we can have a set theory where the axiom of
choice is not allowed and every subset of reals is Lebesgue measurable.
Is this correct?
c.f. Solovay, R.M., ``A model of set-theory in which every set of reals
is Lebesgue measurable.'' Ann. of Math. (2), Vol. 92, 1970. pp. 1--56.