Why are gauge integrals not more popular?

A recent answer reminded me of the gauge integral, which you can read about here.

It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue integrable, it is gauge integrable. (EDIT – as Qiaochu Yuan points out, I should clarify this to mean that the set of Lebesgue integrable functions is a proper subset of gauge integrable functions.)

My question is this: What mathematical properties, if any, make the gauge integral (aka the Henstock–Kurzweil integral) less useful than the Lebesgue or Riemann integrals?

I have just a cursory overview of the properties that make Lebesgue integration more useful than Riemann in certain situations and vice versa. I was wondering if any corresponding overview could be given for the gauge integral, since I don’t quite have the background to tackle textbooks or articles on the subject.

Answer

I would have written this as a comment, but by lack of reputation this has become an answer. Not long ago I’ve posed the same question to a group of analysts and they gave me more or less these answers:

1) The gauge integral is only defined for (subsets of) Rn. It can easily be extended to manifolds but not to a more general class of spaces. It is therefore not of use in (general) harmonic analysis and other fields.

2) It lacks a lot of very nice properties the lebesgue integral has. For example fL1|f|L1 obviously has no generalization to gauge theory.

3) and probably most important. Afaik (also according to wikipedia) there is no known natural topology for the space of gauge integrable functions.

Attribution
Source : Link , Question Author : Chris Brooks , Answer Author : Alexander Thumm

Leave a Comment