The gauge integral is a generalization of the Riemann and Lebesgue integrals. Interestingly, there are functions that can be integrated as improper Riemann integrals but are not Lebesgue integrable. The gauge integral also subsumes improper integrals. Lebesgue integrable functions turn out to be functions f such that both f and |f| are gauge integrable.
The definition of the gauge integral is also much simpler than the Lebesgue integral, as can be seen in this presentation, An Introduction to the Gauge Integral.
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”
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.
Wait. You mean instead of the differential form version??
I mean the differential form version with the weakest possible conditions on the form and the boundary.
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.
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…
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.
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.
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.
Yes there is a cannonical multivariate definition of gauge integral and yes it can be generalized like lebesgue. in fact one can aim that in the lebesgue setting with mild conditions one can assert cousins lemma.
gauge integral is the only perfect solution and the perfect treatment of classical analysis.
it wasw a histrical accidenyt that lebesgue integral was defined first!
agreed that cousin’s lemma is not transparent but then so is the notion of a nonmeasurable set.
How does the canonical multivariate definition work?
Haven’t read it yet (just got it) but:
A General Theory Of Integration In Function Spaces Including Wiener And Feynman Integration
by Muldowney, P. (MR887535 (89i:28006))
is supposed to treat the infinite dimensional case. Has anyone read it?