I’ve tracked down some more papers on the gauge integral, also known as the generalized Riemann integral or Henstock-Kurzweil integral. The Riemann Integral Revisited offers some more details, including a proof that the characteristic function of the rationals is gauge-integrable. Non-Absolute Integrals in the Twentieth Century provides a history of the integral and some extensions.

Several standard theorems have simpler statements using the gauge integral, as shown by the following papers:

• Newton-Leibniz Formula and the Henstock-Kurzweil Integral
• Necessary and Sufficient Conditions for Differentiating Under the Integral Sign
• Stokes’ Theorem