Re: Question Regard Cardinality and Its Implications
From: José Carlos Santos (jcsantos_at_fc.up.pt)
Date: 08/23/04
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Date: Mon, 23 Aug 2004 07:46:10 +0100
Jason Pawloski wrote:
> Most people here are well familiar with the function f(x) = {1 if x is
> rational, 0 is x is irrational } integrates to 0 on any interval. The
> rationals are countable and the irrationals are uncountable, so the
> irrationals are a larger set than the rationals.
>
> From my understanding of the continuum hypothesis, it is conjectured to be
> possible for a set P to have the property that card N < card P < card R. If
> this is the case, would that imply f(x) = { 1 if x is in P, 0 if x is in R \
> P } integrate to non-zero and finite on any non-empty interval? I'm afraid I
> do not know enough about Lebesgue measures to answer the question for
> myself.
That integral is the measure of the set P, which is zero, even without
assuming the continuum hypothesis. Do a Google search for a sci.math
thread called "Continuum hypotheses and Lebesgue measure".
Best regards,
Jose Carlos Santos
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