The gauge/Henstock-Kurzweil integral allows for the integration of a very large set of functions in $\Bbb R$, at the cost of many of the nice properties of Lebesgue integration, of which it is a strict generalization in $\Bbb R$. Essentially, it is a "conditionally convergent" version of Lebesgue integration, where functions that are not absolutely integrable can still be systematically assigned values without defining them as "improper" integrals.

For series, jettisoning absolute convergence yields the Riemann rearrangement theorem where the order of the sequence being summed matters when assigning a value to the sum (if it exists).

Are there any reasonable extensions of the gauge integral that eschew some of the order-independent niceties (Fubini's theorem for one)? Have these any use beyond theoretical niceties?



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