Re: Objective of Riemann Integral??

In article <waderameyxiii-775FAC.22154905042005@xxxxxxxxxxxxxxxxxxxxxxxx>,
The World Wide Wade <waderameyxiii@xxxxxxxxxxxxxxxxxxxx> wrote:
>In article <d2ulkl$se4@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.

>You're being absurd now. Of course it is treated in calculus.
>What planet are you on? The average calc student may not get it,
>but then that student is not going to "consider the spectral
>representation of Hermitian operators on a Hilbert space", LOL.

What proportion of those taking calculus think of the
integral as other than the antiderivative?

They might not think of that particular problem, but
they do think of integrals of complex and vector-valued
functions. and they think of sums, none of which can be
done by considering areas under curves. Not recognizing
that a sum is an integral is definitely a major problem
in understanding of integration.

>> Also, limiting it to functions on an interval, instead of
>> using an arbitrary measure, confuses the issue.

>It does no such thing. At this point I'll leave you to study 200
>pages on the Riemann integral; I'm sure it will give you a great
>one-up on the large population of naive mathematicians who have
>stupidly been brainwashed into buying in to the Lebesgue integral.

Starting out with the usual Lebesgue integral, instead of
integration with respect to an arbitrary measure, often
does result in brainwashing. Also, using the cute tricks
often found in those books adds to more of it, and less
of the needed understanding.
--
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
.

Relevant Pages

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  • Re: Objective of Riemann Integral??
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