Re: The real numbers, and general comments

From: Dave Seaman (dseaman_at_no.such.host)
Date: 09/23/04 

Date: Thu, 23 Sep 2004 12:49:18 +0000 (UTC)

On 22 Sep 2004 23:01:12 -0700, Andrew Usher wrote:
> Dave Seaman <dseaman@no.such.host> wrote in message news:<cirspm$bq$1@mozo.cc.purdue.edu>...

>> >> >> > It's essentially impossible to say anything about a general function.
>> >> >> > Only when you assume at least continuity do you get some theorems.
>>
>> >> >> Look up the Lebesgue Dominated Convergence Theorem.

>> > It does apply to arbitrary functions, but it doesn't give any useful
>> > results, since every function is Lesbesgue integrable.
>>
>> Wrong, and wrong. And you misspelled "Lebesgue".

> Right, it's 'Lebesgue'.

> Every function is Lebesgue integrable. (Actually, the integral might
> be infinite but this nullifies the convergence theorem.)

> Proof:

> Let f(x) be any function and consider the set {x : f(x) >= a}. This
> set is constructible (we just did), so it is measurable. Since every
> such set has a measure, f(x) is measurable. All measurable functions
> have a Lebesgue integral.
> Q.E.D.

The theorem says something more than just that a certain function is
measurable. Even if you restrict your attention to the constructible
universe (which I don't), the theorem has content.

For one thing, the LDCT offers yet another easy proof that the reals are
uncountable. Do you consider that uninteresting? 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

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