Re: Objective of Riemann Integral??

In article <waderameyxiii-64B3CE.20451931032005@xxxxxxxxxxxxxxxxxxxxxxxx>,
The World Wide Wade <waderameyxiii@xxxxxxxxxxxxxxxxxxxx> wrote:
>In article <d2h388$1g56@xxxxxxxxxxxxxxxxxxxx>,
> hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin) wrote:

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

...................

>> 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.

>But the idea of the integral as a limit of sums is treated in
>first semester calculus. Everyone knows it; everyone understands
>it. What Dieudonne is questioning is a 200 page treatise devoted
>to Riemann integration. I'll bet a lot of that book is devoted to
>higher dimensions. What's the point? Without the Lebesgue
>integral that's a tortuous road, replete with all manner of
>needless worries that the Lebesgue integral steamrolls over
>cleanly.

The integral as a limit of sums is MENTIONED in calculus.
It is not TREATED. Few of the students get that idea.

Also, limiting it to functions on an interval, instead of
using an arbitrary measure, confuses the issue. One would
do better with starting out with number of grade points
as the integral of the numerical value of the grade with
respect to the measure of a set of courses being the total
number of credits, or computing a bill.
--
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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