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From: ags@pucc-i (Dave Seaman)
Newsgroups: net.math
Subject: Re: f(x) = (if x = p/q then 1/q else 0) integrable ??
Message-ID: <869@pucc-i>
Date: Tue, 29-Jan-85 09:51:26 EST
Article-I.D.: pucc-i.869
Posted: Tue Jan 29 09:51:26 1985
Date-Received: Thu, 31-Jan-85 01:57:43 EST
References: <350@decwrl.UUCP>
Organization: Purdue University Computing Center
Lines: 14

>  What is the integral ?  Or what is it from x = 0 to 1 ?

The answer to both questions is zero.  The function is zero "almost
everywhere," which means everywhere except on a set of measure zero.
The rational numbers are a countable set, and all countable sets have
measure zero.  Whenever two functions are equal almost everywhere, it 
follows that they have the same integral.

The integral I am referring to is the Lebesgue integral, which is not the
one taught in beginning calculus courses.  The older Riemann integral does
not exist for the function f.  Whenever a function is Riemann-integrable,
it is also Lebesgue-integrable, and the integrals agree.
-- 
Dave Seaman			 ..!pur-ee!pucc-i:ags