Re: Why do they teach Riemannian sums?

Jiri Lebl writes:
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.

It is arguable that such functions aren't really integrable, since
their Lebesgue integral diverges. The convergence is only because of
the restriction on the limiting process inherent in the naive
definition of the improper Riemann integral.

For example, the function f(x) that is equal to (-1^k)/k on each
interval [k,k+1) (for positive integer k) is such that the limit as b
goes to infinity of int^b_0 f(x) dx exists, and thus the naive improper
Riemann integral exists. But clearly there are integrals over
subsets of the domain that diverge, so monotonicity fails. Whenever I
define improper integration in the introductory calculus classes I
teach, I always add the restriction that both the integral of the
positive part and the integral of the negative part must exist before
we can say that the function is integrable. Or I tell the students we
will only talk about nonnegative functions, if I am feeling lazy.

I would say the only reason that my definition isn't part of the
textbook definition is that nobody cares about Riemann integrals any
more, so they are frozen with the definitions they had in 1880. In my
opinion, monotonicity is too important a property of integrability to
say that functions such as f(x) above are integrable. Whether the
naive improper Riemann integral is useful for the physical sciences is
another matter.

.

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