Re: Why do they teach Riemannian sums?

Jiri Lebl writes
JL>OK, I will quote you from just a post ago since you've obviously
JL>forgotten what you said:
JL>M> "The limiting nonsense just gets worse with HK (i.e. gauge)
integrals.

I took the word nonsense directly from Lebl's previous post, from which
I quote:
JL>But such functions are HK integrable and there this limitting
JL>nonsense with riemann integrals disappears.

In a more recent post, Libl goes on to say:
You are now arguing like Tony Orlow by making up your definition of what it
means to be "integrable" rather then taking the actual standard definition.

There is no ``standard'' definition, except possible that of Lebesgue
integration.
Certainly, in very few places does the bare word ``integrable'' mean
anything except a member of L^1. I am sure that this is what
Halmos's and Rudin's books, from which I learned integration, say.
Very few people use any other definition of the word.

It is clear from my posts that I am trying to talk about about what the
word ``integrable'' ought to mean in some philosophical sense. I
understand the definitions of Riemann, Lebesgue, and HK integration and
I have no difficulty with them. I do not argue that these definitions
are not standard, well known, and useful. I have even said (twice)
that I am not arguing whether different integrals are useful for the
physical sciences.

Please look through my previous posts and you will see that I have
never said that the gauge integral is not adequately defined, useless,
etc. I only said it doesn't match my idea about what integrability
ought to be.

Libl says:
I know of NO analysis book defining an integral as an area under the graph.

Determing the area between two curves, or between a curve and the x
axis, has been the standard motivation for integration since it was
developed. I have read that there was once disupute over whether the
area under a parabola was even well defined, because the Greek could
not compute it. The initial motivation for integral calculus was to
solve this kind of problem.

The fact that the integral of a function happens to be related to the
antiderivative of the function is so surprising that it is regarded as
the fundamental theorem of calculus. Antiderivatives are certainly not
the reason that integrals were developed.

I dont have any analysis books at home with me, I know it's lame, but
let me quote from Wikipedia, the first line of the article on
``Integral'':

WP>In calculus, the integral of a function is a generalization of area,

WP> mass, volume, sum, and total.

Note that the first word there is ``area,''

Or actually you want some "area" idea to ALWAYS be tied to the word "integrable".

Yes, now you see what I am saying. The original motivation of the
integral was to find the area between curves. If you want integral to
relate to something else, please tell me what intuition underlies the
definition of the Riemann, HK, or Lebesgue integral of a real-valued
function of a real variable except the idea of determining an area.

To be perfectly clear: I am trying to discuss what the word
``integrable'' ought to mean, since there are several different
integrals that have different collections of integrable functions. My
original comment on the whole subject was:

It is arguable that such functions aren't really integrable, since
their Lebesgue integral diverges.

From that quote, it ought to be clear that I was planning to say that
Lebesgue integrability represents ``real'' integrability.

.

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