3 ms·
I don't think the distinction between the signed and unsigned integral exists for the most general integral, the Henstock-Kurzweil integral. (I could be wrong,
by BlackFly 6y ago
I don't think the distinction between the signed and unsigned integral exists for the most general integral, the Henstock-Kurzweil integral. (I could be wrong, but an orientation seems to always be implied in being able to compute a Riemann sum over a tagged partition.)
This distinction is probably related to the Lebesgue integral's inability to integrate functions unless they are absolutely integrable (since it needs to be able to compute the positive and negative components and take the difference, which being finite to make sense, must be absolutely finite) and is distinct from the Henstock-Kurzweil integral which works directly with a tagged partition of the underlying set (which implies an orientation) and is able to integrate (some) functions which are not absolutely integrable.
Nevertheless, moving to differential manifolds introduces problems for the Henstock-Kurzweil integral becames the local orientation in charts does not always induce a sensible global orientation. However, integration on a manifold isn't so much integration of a function of several variables as it is integration of several functions of several variables. That one needs additional machinery to deal with the "several functions" part is unsurprising.
I can't recommend studying the Henstock-Kurzweil integral enough! Strangely enough, despite being more general, it is far more approachable than the Lebesgue integral.
- mehrdadn 6y agoWikipedia says there's an even more general integral (https://en.wikipedia.org/wiki/Khinchin_integral https://en.wikipedia.org/wiki/Khinchin_integral)? https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_inte...
- BlackFly 6y agoGeneral in the sense that it integrates more functions. The Denjoy integral is equivalent to the Henstock-Kurzweil integral in that respect (Theories of Integration - The integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane, Kurtz & Swartz).
- jordigh 6y agoI'm not sure gauge integrals are really "the most general integral". As you say, it doesn't work as well as the Lebesgue integral in multidimensional settings, does it? It also needs a bit of a tweak to give you the analogue of Stieltjes integration so you can unify sums and integrals, I believe.
- BlackFly 6y agoThey are the most general in the sense that there are Henstock-Kurzweil integrable functions that are not Lebesgue integrable and that other integrals that are also more general than the Lebesgue integrals are equivalent to the Henstock-Kurzweil integral. It still works better than the Lebesgue integral in the multidimensional settings, since it is trivial to create a product f(x)g(y) of two functions which will not be Lebesgue integrable but is Henstock-Kurzweil integrable. As for generalizations to generalized functions, my preference lies with Colombeau algebras over Schwartz distributions, in any case. Where at least there is an arithmetic of the generalized functions.