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 f∈L1⇒|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*