A reasonable method of defining an integral that includes the HK integral is to say a Schwartz distribution f is integrable if it is the distributional derivative of a continuous function F. Then the integral ∫_a^b f =F(b) - F(a). The resulting space of integrable distributions is a Banach space that includes the space of HK integrable functions and is isometrically isomorphic (with Alexiewicz norm) to the continuous functions vanishing at a (with uniform norm). In Euclidean spaces one does much the same but integrable distributions are those that are the distributional derivative taken once in each Cartesian variable. Again, we get a Banach space of integrable distributions. A Fubini theorem holds but the only transformations that can be done are those that map Cartesian intervals to Cartesian intervals. This rules out rotations. For details, see D.D. Ang and L.K. Vy, {\it On the Denjoy--Perron--Henstock--Kurzweil integral}, Vietnam J. Math. {\bf 31}(2003), 381--389.

A reasonable method of defining an integral that includes the HK integral is to say a Schwartz distribution $f$ is integrable if it is the distributional derivative of a continuous function $F$. Then the integral $\int_a^b f =F(b) - F(a)$. The resulting space of integrable distributions is a Banach space that includes the space of HK integrable functions and is isometrically isomorphic (with Alexiewicz norm) to the continuous functions vanishing at a (with uniform norm). In Euclidean spaces one does much the same but integrable distributions are those that are the distributional derivative taken once in each Cartesian variable. Again, we get a Banach space of integrable distributions. A Fubini theorem holds but the only transformations that can be done are those that map Cartesian intervals to Cartesian intervals. This rules out rotations. For details, see D.D. Ang and L.K. Vy, On the Denjoy--Perron--Henstock--Kurzweil integral, Vietnam J. Math. 31 (2003), 381--389.