Well if (i) Lebesgue integrable implies measurable and (ii) every gauge integrable function is measurable and (iii) ##f## is Lebesgue integrable iff ##f## is measurable and ##|f|## is gauge integrable" (three assertions taken from your posts) then my statement follows.

The Insight article has nice pictures that give intuition about the Riemann integral and the Legesgue integral. Is there a useful picture that explains the gauge integral ?

The arguments given in favor of the gauge integral focus on the nice implications it provides - if f is gauge integrable then ..... To use such implications in a specific setting, one would need to establish the "f is gauge integrable" clause. I don't get any intuitive understanding of how to do that, except in a trivial case where we can define [itex] \delta(x) [/itex] to be a constant function, reverting it to the ordinary [itex] \delta [/itex].

For example, I gather that [itex] \int_0 ^ {\infty} \frac{\sin{x}}{x} [/itex] can be defined as an "extended" gauge integral in the usual way, by taking the limit of a gauge integrals over a finite intervals. So, technically, we need the existence of a different function [itex] \delta(x) [/itex] for each of the finite intervals.

Does the "gauge" in gauge integral has something to do with the concept of "gauge" in physics?

Interesting. So do you as for improper Riemann integrals define for example the gauge integral ##\int^{\infty}_a f(x) dx## as the limit of the gauge integrals ##\lim_{t \to \infty} \int^t_a f(x) dx## (whenever the limit exists or is ##\pm \infty##) where ##f : [a, \infty) \to \mathbb{R}## is gauge integrable on ##[a,t]## for all ##t \geq a##? As far as I can see, you only gave a definition of a gauge integrable function on a closed interval.

Assuming the above, what's stopping us from simply defining ##\int^{\infty}_a f(x) dx## as the limit of the Lebesgue integrals ##\lim_{t \to \infty} \int^t_a f(x) dx## (whenever the limit exists or is ##\pm \infty##) for functions ##f : [a,\infty) \to \mathbb{R}## which are Lebesgue integrable on ##[a,t]## for all ##t \geq a##? Using the same idea for all the different improper integrals would give us a notion of improper Lebesgue integrals, and improperly Lebesgue integrable functions.

It seems to me that perhaps the theorem you gave about the equivalence between Lebesgue integrable functions and gauge integrable functions such that ##\int |f| < \infty## implies that the definition above of a Lebesgue integrable function (proper and improper) is equivalent with the definition of a gauge integrable function.

Maybe I'm wrong and missing something here. What exactly can gauge integration do which Lebesgue integration and improper Lebesgue integration like I defined above cannot?

Also, what do you do about gauge integrals ##\int_A f(x) dx## for arbitrary sets ##A##? For which sets and functions are such integrals defined?