Comments on: Gauge integral

By: Jacob Freeze

Jacob Freeze — Mon, 17 Mar 2008 17:14:23 +0000

As well as I can understand the Denjoy-Perron-Henstock-Kurzweil integral (not very), the advantages for improper integrals over intervals may be outweighed by problems generalizing the definition to cover infinite dimensional spaces.

It would still be nice to dispense with Lebesgue’s sigma algebras, but the delta functions that define a gauge aren’t exactly transparent, either… at least, not to me.

Apparently the gauge crusaders also have a little trouble selling the extra machinery for the definition (Cousin’s Lemma) to Calc 101 teachers.

By: Walt

Walt — Wed, 12 Mar 2008 04:08:58 +0000

Is there a clear and canonical multivariate version of the Henstock integral? My impression (which could be wrong) is that there are multiple definitions, none of which is particularly elegant.

By: prof anil pedgaonkar

prof anil pedgaonkar — Tue, 11 Mar 2008 17:18:03 +0000

Henstock integral is really the complete notion of integral on domin of euclidean spaces. it ofers the complete treatment of calculus especially in banach spaces. contrary to opinion this can be genralized locally compact topological groups and locally compact spaces as wel l and a general approach like that of lebesgue is also posiible.
our undergraduate and graduate analysis courese nedd to be redesigned. lebesgue integral is nothing but absolute henstock integrability. dieodenne and lang’s tratments of amnlysis in banach spaces need to be covered rather than sticking to euclidean spaces.
surely it is henstock integral which will dominate the scene in 21 st century. lebesgue is outdated. measure of a set is something which one computes by integration and not then other way round as in lebesgue theory an unfortunate accident in history. please do contact me on mail for further discussion and your views.

By: oboe

oboe — Sat, 29 Dec 2007 13:26:22 +0000

somewhere along the road, the lecturer forgot to taught what is an integral? what is that we are trying to capture with that concept?nevermind…

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”