NetNews Usenet Archive 1992 #20/NN_1992_20.iso/spool/sci/math/11184

Internet Message Format | 1992-09-09 | 1.8 KB

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From: [e-mail] (Allan Adler)
Newsgroups: sci.math
Subject: Re: Couple of questions
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Date: 10 Sep 92 01:25:30 GMT
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In-Reply-To: [e-mail]'s message of 9 Sep 92 17:49:10 GMT

There is now a generalization of the Riemann integeral, called the Generalized
Riemann Integral, which is powerful enough to integrate all Lebesgue integrable
functions. A positive function is integrable by this function if and only
if it is Lebesgue integrable. However, there are functions which are not
Lebesgue intergrable, by virtue of the fact that their absolute values are
not, but which are integrable by the Generalized Riemann Integral. There is
a Carus monograph on this. I think McShane has a book that handles the
Generalized Riemann Integral in R^n and for for the infinite dimensional
integrals that arise in stochastic processes. There is a very recent book
by Henstock which seems to achieve the greatest generality, axiomatizing
the structures necessary to carry out constructions of this type.

There are some people who work in integration theory who believe that the
Lebesgue integral is now obsolete and that the generalized Riemann integral,
including its general formulation, should be the foundation that is taught.

I think the generalized Riemann integral does one other thing: every
differentiable function is the generalized Riemann integral of its
derivative.

Allan