Re: Objective of Riemann Integral??

In article <waderameyxiii-F1B7AE.10195730032005@xxxxxxxxxxxxxxxxxxxxxxxx>,
The World Wide Wade <waderameyxiii@xxxxxxxxxxxxxxxxxxxx> wrote:
>In article <d2bs0b$1nd2@xxxxxxxxxxxxxxxxxxxx>,
> hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin) wrote:

>> >As a general comment, the reviewer thinks one may well question the
>> >advisability of devoting so much labor and care to the exposition of a
>> >theory
>> >which, half a century after Lebesgue's thesis, appears more and more as a
>> >mathematical curiosity. Why should a student bother to read 200 pages on the
>> >Riemann integral when any standard book on the Lebesgue integral will give
>> >him
>> >in half that length a much clearer insight into the nature of the integral,
>> >with no appreciable increase in the complexity of the proofs?
>> > Reviewed by J. Dieudonne

>> Dieudonne, and it seems the entire Bourbaki school, seemed to
>> have blinders on when it came to measure and integration.
>> There is a need for non-Lebesgue integration, even in analysis,
>> such as the Hilbert transform. Integrals where the limiting
>> process is not the Lebesgue type, with integrands such as
>> sin(x)/x, are of considerable importance.

>Those are non-examples in my opinion. You need to use limiting
>processes in both cases that are extensions of the ordinary
>integral. As the Riemann integral is a special case of the
>Lebesgue integral, there is no example where the Riemann integral
>works and the Lebesgue integral doesn't.

The Riemann integral might happen to coincide with the
Lebesgue integral for those functions on a finite
interval for which the Riemann integral exists. However,
the idea of the Riemann integral is NOT a special case
of the idea of the Lebesgue integral, as countable
additivity is not required for the Riemann extension.

One could use the definition of the Riemann integral for
functions defined only on the rationals; in this case,
the Lebesgue extension does not work.

>> In the field of
>> stochastic integration, even Riemann can be "too much",
>> and limits in probability are needed. Riemann-Stieltjes
>> and Riemann-Hellinger integrals are also useful.

>But Dieudonne is talking about Riemann vs. Lebesgue; he's not
>not denying the existence of situations where neither is enough.
>(And anyway, Riemann-Stieltjes integrals are just a special
>little area of measure theory.)

Stochastic Riemann-Stieltjes integrals can be used in
cases in which a countably additive measure structure
does not exist; they are useful, as are similar
Riemann-Hellinger integrals. These are not even as
strong as Riemann integrals.

Also, if one wants such things as line integrals, or
complex integrals, including path integrals, or integrals
of function in linear spaces, or product integrals, only
a limit approach can be used, and it works well. Area
under a curve only works for the integral in one dimension
with length measure.

One can even use the limit formulation to consider the
spectral representation of Hermitian operators on a
Hilbert space, with the measure being projection valued.
It is still the limit of finite sums, appropriately
defined.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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