Re: Measure and countability: question
From: Robert Israel (israel_at_math.ubc.ca)
Date: 01/07/05
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Date: 7 Jan 2005 03:27:02 GMT
In article <de044e77.0501061901.3626ff36@posting.google.com>,
tetrahedron <jarynth@yahoo.com> wrote:
> Let's talk about the sigma-algebra of Lebesgue-measurable subsets of
>the reals. What changes if in the axioms for a sigma-algebra we allow
>for the uncountability of unions?
Every set is a union of singletons, so a sigma-algebra containing all
singletons and closed under arbitrary unions consists of all subsets
of the base set. Assuming the Axiom of Choice, some of these are
non-measurable.
> In conclusion, is there another way of showing that any countable
>subset of the reals, e.g. the rationals, is negligibly dense in R,
>aside from measure theory?
Any countable subset is also negligible (technical terms are "meagre" or
"first category") in the sense of the Baire Category Theorem.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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