Re: question: Lebesgue measure

From: Ron Sperber (ronsperber_at_optonline.net)
Date: 12/31/04 

Date: Thu, 30 Dec 2004 19:08:57 -0500

tetrahedron wrote:
> I was wondering how many nonmeasurable sets exist (e.g. as subsets of
> the reals). Since AC is needed to prove their existence, is it
> possible to make any statement about their number at all?
>
> Second, I don't understand why the Cantor set is uncountable. It's not
> even dense. Why am I wrong?
>
> Best regards

I'm not sure as to the source of your confusion. Are you under the
impression that an uncountable set must be dense? Dense in what?
The interval [0,1] is uncountable and not dense in the real line.
As to why, there are a few approaches. One is to show that the cantor
set is perfect (i.e. equal to the set of its limit points) and then show
that perfect sets are uncountable. Another is to show that the Cantor
set is homeomorphic to the countable product of copies of {0,1} which
can't be countable.

Relevant Pages

  • Re: DSP riddle
    ... the rational numbers lie dense in the real numbers (every ... The mathematician who did it was called Cantor ("triangulation of Cantor" is the name of the proof that there are reals which are not rationals) and he became mad after that ...
    (comp.dsp)
  • Re: First diagnal proof for real numbers
    ... expansions of real numbers and diagonalization. ... already Russell in 1903, contrary to Cantor, explicitly considers ... the non-uniqueness of expansions, he was well aware of it. ... of constructing the reals from the rationals by the use ...
    (sci.math.research)
  • Re: A wise decision
    ... Cantor showed how infinite sets could have different cardinality, ... i.e. irrationals like e.g. pi as well as the "other" reals. ... i.e. mistakes already at the level of mathematics. ...
    (sci.math)
  • Re: Scattered sets are G-delta
    ... Sets with countable closure ... Take any Cantor set (perfect nowhere dense ... from the reals). ...
    (sci.math)
  • Re: Existence of reals and observation of them
    ... Cantor says that the set of all finitely definable reals is ... > definitions of reals. ... > It's important that he never refers to an uncountable set of symbols ... infinite sets we can finitely define. ...
    (sci.math)