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Re: Lebesgue Measurable set that is not Borel?
Posted:
Jul 31, 2012 8:12 AM


In article <299fb76cbdcc4a9e83509554f0e00133@googlegroups.com>, <baclesback@gmail.com> wrote:
> 2)Anyone know of an actual construction of this set? > > Thanks.
I posted in this very newsgroup October 1993, and reposted several times since:
> > An explicit example of a set of real numbers that is measurable (indeed, > analytic) but not Borel [due to Lusin, Fundamenta Math. 10 (1927) p. > 77]: > > the set of all real numbers x with continued fraction expansion > > x = a[0] + 1/(a[1] + 1/(...)) > > such that, for some positive integers r[1] < r[2] < ..., we have > a[r[i]] divides a[r[i+1]] for all i. > > Other examples of analytic sets that are not Borel can be given > in (complete separable) metric spaces other than the line: > > In the space K[0,1] of nonempty compact subsets of [0,1] with > the Hausdorff metric: The subset consists of the uncountable > compact subsets. [Hurewicz, 1930] > > In the space C[0,1] of realvalued continuous functions on [0,1] > with the unform metric: The subset consists of the differentiable > functions. [Mazurkiewicz, 1936] >
 G. A. Edgar http://www.math.ohiostate.edu/~edgar/



