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.