Re: Numerical methods for Lebesgue integrals?

In article <RzZXg.92$3C6.30@trnddc04>, Ben Crain <bcrain@xxxxxxxxxxx> wrote:
Lebesgue extends Riemann. Are there numerical methods for approximating
those Lebesgue integrals for which Riemann doesn't exist? I get the
impression that issue may never arise, in practice. From my (quite limited)
acquaintance with Lebesgue, Henri seems to be used primarily in theoretical
work, where an actual numerical evaluation/approximation is not needed. Is
that always true? Does it never arise that a Lebesgue integral needs to be
actually evaluated, and there is no Riemann equivalent? A related question:
Even when Riemann exists, so Lebesgue = Riemann, could numerical techniques
applied to the Lebesgue formulation of the integral be developed which might
be competitive with, or even superior to Riemann techniques? Maybe they
have been, and I just haven't yet found them.

Practical examples would include functions on infinite intervals, or
functions with singularities. From the Riemann point of view these
would typically be handled as improper integrals, using various
techniques. I don't know of any specifically "Lebesgue" techniques for
numerical computation in general.

Well, you might consider something like this: suppose you define

f(x) = sum_{n=1}^infty f_n(x)

where you happen to know that sum_{n=1}^infty int_0^1 |f_n(x)| dx
is finite, but you don't have good pointwise bounds: in particular,
the series does not converge uniformly, and might diverge at some
points. You wouldn't even know if f is Riemann integrable. You
need the Lebesgue theory to tell you that

int_0^1 f(x) dx = sum_{n=1}^infty int_0^1 f_n(x) dx

and then you might apply numerical methods to calculating this
sum.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

.

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