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From: Maury Barbato on 5 May 2010 08:47
Hello,
let R be the set of real numbers. Let A, b be two
subsets of R and f:A-> B a homeomorphism. If A
is Lebesgue measurable, is also B Lebesgue mesurable?

I don't if the question is trivial or not, because
I know very little about Lebesgue measure theory.
Thank you very much for your attention.
My Best Regards,
Maury Barbato
From: cwldoc on 5 May 2010 09:16
> Hello,
> let R be the set of real numbers. Let A, b be two
> subsets of R and f:A-> B a homeomorphism. If A
> is Lebesgue measurable, is also B Lebesgue
> mesurable?
>
> I don't if the question is trivial or not, because
> I know very little about Lebesgue measure theory.
> Thank you very much for your attention.
> My Best Regards,
> Maury Barbato


I'm not sure, but I think the following is true:

Since f continuous, it is also a measurable function. If B is measurable, then
A = [preimage of B] is measurable. Similarly, if A is measurable, then so is B.
From: Robert Israel on 5 May 2010 14:07

> > Hello,
> > let R be the set of real numbers. Let A, b be two
> > subsets of R and f:A-> B a homeomorphism. If A
> > is Lebesgue measurable, is also B Lebesgue
> > mesurable?
> >
> > I don't if the question is trivial or not, because
> > I know very little about Lebesgue measure theory.
> > Thank you very much for your attention.
> > My Best Regards,
> > Maury Barbato
>
>
> I'm not sure, but I think the following is true:
>
> Since f continuous, it is also a measurable function. If B is measurable,
> then
> A = [preimage of B] is measurable. Similarly, if A is measurable, then so
> is B.

No, this is not true. What is true is that A is Borel iff B is Borel.
However, let C be the usual Cantor set of measure 0, and f a
homeomorphism of C onto a "fat Cantor set" D of positive measure.
Let B be a nonmeasurable subset of D (these exist: e.g. the intersection
of D with a Bernstein set). But A = f^(-1)(B) is Lebesgue measurable
(because it has measure 0).
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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