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Topic: Lebesgue- Measurable set that is not Borel?
Replies: 1   Last Post: Jul 31, 2012 8:12 AM

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 G. A. Edgar Posts: 2,510 Registered: 12/8/04
Re: Lebesgue- Measurable set that is not Borel?
Posted: Jul 31, 2012 8:12 AM

<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 real-valued continuous functions on [0,1]
> with the unform metric: The subset consists of the differentiable
> functions. [Mazurkiewicz, 1936]
>

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

Date Subject Author
7/31/12 Bacle H
7/31/12 G. A. Edgar