Re: Why do they teach Riemannian sums?

Timothy Murphy wrote:
Jiri Lebl wrote:

Riemann sums are thaught so that one UNDERSTANDS the theory. It turns
out that understanding what the hell is going on is far supperior to
memorizing billions of little facts with little or no understanding.

I'm not sure what "Riemann sums" are,
but I think the teaching of Riemann integration is completely misconceived,
especially where students are later taught Lebesgue integration.

Since I've seen first hand how much trouble Riemann integration gives
to undergrad analysis students, I would not try anything more then
Riemann-Stieltjes on them and expect them to absorb the material.

Also if we are talking about calculus students. Then it sometimes
seems even Riemann integration is too tough.

In any case, if you use Riemann sums you see very easily why
integration is really a generalized summation. It is much more
concrete then Lebesgue integration. Think about how much background
you have to cover before you get to give a definition of a Lebesgue
integral.

Further, contrary to popular belief, not all Riemann integrable
functions (in the improper riemann integral sense) are Lebesgue
integrable. Things that oscillate wildly near a point (or near
infinity) may have an improper Riemann integral and NOT a Lebesgue
integral.

So perhaps if you want the most general integral while staying within
R^k and keeping the material concrete enough for beginning analysis
students to understand, perhaps the Henstock-Kurzweil is the best bet.
Plus, if they have seen Riemann integral, HK is a small generalization
of the definition, while it mops up ALL the functions on R^k you can
possibly wish to integrate (including some non-lebesgue integrable
ones).

Jiri

.

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