Re: The real numbers, and general comments

From: Andrew Usher (k_over_hbarc_at_yahoo.com)
Date: 09/23/04 

Date: 22 Sep 2004 23:01:12 -0700

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.

Andrew Usher

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