Slick proofs using the Henstock–Kurzweil integral?

Question

I enjoyed Iosif Pinelis's slick answer to another MO problem using the Henstock–Kurzweil integral. Are there other examples of problems whose statement does not explicitly involve the Henstock–Kurzweil integral, but which can be solved more elegantly using the Henstock–Kurzweil integral than by more conventional methods?

I'm sympathetic to the notion that the Henstock–Kurzweil integral should be more widely taught as a theory of integration (at least in dimension one), but I'd like to see more examples of its usefulness.

Answer 1

My Ph.D. thesis (over 340 pages), Henstock–Kurzweil Integral in Abstract Set Up,, conducts treatment of Fubini theorem. Also Henstock–Kurzweil integral right from the beginning is treated on $\mathbb{R} = [-\infty, \infty]$ and $\mathbb{R}^n$, and now I also have $\mathbb{R}^s$ where $s$ can be countable or uncountable infinite. It carries the Riemmann sum convergence theorem and generalized Lebesgue dominated convergence. In fact now I have a generalized Fatou's lemma for arbitrary functions and a generalized monotone convergence theorem mean value theorem which gives as a corollary Lagrange's mean value theorem. My thesis can be downloaded from UGC india website. You can get it by email from me; mail [email protected] The thesis is refereed by Muldowney, a student of Henstock.