3 ms·

I'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,

I'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 ago
They 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.