Re: Objective of Riemann Integral??

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.

> 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.
.

Relevant Pages

  • Re: Positive/Negative after taking the square root
    ... With the substitution u = x^2 this appears to become an integral ... Suppose we wish to calculate the average distance of a planet from a star, ... Method 2 (for the industrious student who eschews tables or CASs): ... seen to be wrong because the limits of integration, t=0 to t=2 pi, become ...
    (sci.math)
  • Re: Objective of Riemann Integral??
    ... >> The integral as a limit of sums is MENTIONED in calculus. ... The average calc student may not get it, ... in understanding of integration. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math)