Re: Why do they teach Riemannian sums?

I would say that the HK integral is not ``more general'' than the
Lebesgue integral, because there are properties of the Lebesgue
integral that the HK integral does not share. Monotonicity, for
example. It is true that more functions are integrable in the HK
integral than in the Lebesgue integral, but the fact that certain
functions are not integrable is an important property of the Lebesgue
integral. Other people are free to argue ``if it integrates more
functions it's more general'' but I politely disagree about the meaning
of the words ``more general.''

Jiri Lebl writes
A rapidly oscillating function need not have an "area" under it's graph.

Then you are throwing out:

(2) The integral of a function is the area under its graph.

if you call that function integrable. In my conception of integration
of real-valued functions on the real line, the integral is exactly the
signed area determined by the graph of the function and the x axis.
One cannot exist without the other - if the area is not well defined
then the function must not be integrable. Put another way, the
integrable functions are all examples of functions for which the area
is well defined. The basic motivation of the theory of integration
is to determine a large class of functions for which the area is well
defined.

The only place where you get a function which is not lebesgue but is HK
integrable is if you allow negative values, in which case you're
talking about some "signed area" which really only makes sense if both
positive and negative parts are finite.

That's exactly my point: it makes no sense to speak of ``integrable''
functions in which the negative part and the positive part are both
nonintegrable.

.

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