Comments on: Gauge integral

By: artem

artem — Sun, 28 Jan 2007 18:04:58 +0000

I’ve studied it in the first year of my calculus course, together with the Riemann integral and McShane version of the Lebesque integral. Nice experience … though I didn’t get an idea what is this enormous power good for.

By: Walt

Walt — Fri, 24 Jun 2005 17:20:01 +0000

I mean the differential form version with the weakest possible conditions on the form and the boundary.

By: michael

michael — Fri, 24 Jun 2005 09:11:29 +0000

Wait. You mean instead of the differential form version??

By: Walt

Walt — Fri, 24 Jun 2005 08:32:39 +0000

I think this is a case where a concept has two different generalizations and both are interesting. The Lebesgue integral can be defined for arbitrary sets, and allows you to define normed vector spaces. The gauge integral is restricted to the real line, but it allows you to compute oscillatory integrals, and satisfies the strongest possible analogue of the Fundamental Theorem of Calculus: the gauge integral of f’ is always defined, and equals f. I suspect that the proper setting for something like Stokes’ theorem is a higher-dimensional analogue of the gauge integral.

By: michael

michael — Thu, 23 Jun 2005 20:59:25 +0000

I read this and said to myself “That sounds like the Kurzweil-Henstock Integral”, and sho ’nuff. I asked Folland once why we were taught Lebesgue instead, and his reply was “It doesn’t generalize to other spaces” which I guess means “i.e. you will eventually have to learn Lebesgue anyway”