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This is referring to question 1.

There's a choice we have to make when defining a real-valued two-dimensional (or n dimensional) nonabsolute integral: would we rather have a class of integrable functions which include all divergences of differentiable functions, or would we want to have some sort of Fubini's Theorem working? This conflict was know from the early development of the HK integral and was also pointed out by Pfeffer in the book mentioned in Gerald Edgar's answer.

If we define the HK integral in the obvious way (call it the standard two dimensional HK integral), we get Fubini's theorem, but no fully general divergence theorem. Many authors made modifications on the definition of this integral, and an relatively satisfactory definition was given by Jarník, Kurzweil and Schwabik in "On Mawhin's approach to multiple nonabsolutely convergent integral", Casopis. Pest. Mat. There they defined the $M_1$-integral, which satisfies a fully general divergence theorem, and has a simple enough definition so we can prove some convergence theorems. It is shown though that this integral does not satisfy Fubini's theorem when the corresponding one-dimensional integrals are considered to be the HK integral. The original example from that paper can be used to show that, in a more general setting, an interval-based two-dimensional integral that satisfies a full divergence theorem will fail in some sense Fubini's theorem (see Proposição 2.5 here if you are not afraid of reading in portuguese).

Other problem that is frequently overlooked when defining interval-based two-dimensional nonabsolute integrals is that the integral can be sensitive to rotations, that is, we can get integrable functions such that a certain rotation of that function is not integrable. We have this unpleasant effect for the $M_1$-integral and even for the standard two-dimensional HK integral (see main theorem of "K teorii vícerozmerného integrálu", Casopis. Pest. Mat. by K. Kartak, if you are not afraid of reading czech, or Proposição 1.7 in the aforementioned Thesis, which is for the $M_1$-integral but easily adaptable to the standard two-dimensional HK one).

Then there is a new challenge: trying to define an integral which is not based on intervals but that still will be simple enough to prove convergence theorems. Kurzweil himself defined an integral where the domain is partitioned into sets with boundaries continuously differentiable by parts; it's a lot of trouble even to prove Saks' Lemma for this integral. See also this article for an integral where we use triangular partitions. This integral satisfies many of the commonly desired theorems, but it is unknown to me for example if it satisfies a nice change of variables formula.

This is referring to question 1.

There's a choice we have to make when defining a real-valued two-dimensional (or n dimensional) nonabsolute integral: would we rather have a class of integrable functions which include all divergences of differentiable functions, or would we want to have some sort of Fubini's Theorem working? This conflict was know from the early development of the HK integral and was also pointed out by Pfeffer in the book mentioned in Gerald Edgar's answer.

If we define the HK integral in the obvious way (call it the standard two dimensional HK integral), we get Fubini's theorem, but no fully general divergence theorem. Many authors made modifications on the definition of this integral, and an relatively satisfactory definition was given by Jarník, Kurzweil and Schwabik in "On Mawhin's approach to multiple nonabsolutely convergent integral", Casopis. Pest. Mat. There they defined the $M_1$-integral, which satisfies a fully general divergence theorem, and has a simple enough definition so we can prove some convergence theorems. It is shown though that this integral does not satisfy Fubini's theorem when the corresponding one-dimensional integrals are considered to be the HK integral. The original example from that paper can be used to show that, in a more general setting, an interval-based two-dimensional integral that satisfies a full divergence theorem will fail in some sense Fubini's theorem (see Proposição 2.5 here if you are not afraid of reading in portuguese).

Other problem that is frequently overlooked when defining interval-based two-dimensional nonabsolute integrals is that the integral can be sensitive to rotations, that is, we can get integrable functions such that a certain rotation of that function is not integrable. We have this unpleasant effect for the $M_1$-integral and even for the standard two-dimensional HK integral (see main theorem of "K theorii vícerozmerného integrálu", Casopis. Pest. Mat. by K. Kartak, if you are not afraid of reading Czech, or Proposição 1.7 in the aforementioned Thesis, which is for the $M_1$-integral but easily adaptable to the standard two-dimensional HK one).

Then there is a new challenge: trying to define an integral which is not based on intervals but that still will be simple enough to prove convergence theorems. Kurzweil himself defined an integral where the domain is partitioned into sets with boundaries continuously differentiable by parts; it's a lot of trouble even to prove Saks' Lemma for this integral. See also this article (Triangle Integral–A Nonabsolute Integration Process Suitable for Piecewise Linear Surfaces by Ricardo Bianconi and Pedro L. Kaufmann, projecteuclid) for an integral where we use triangular partitions. This integral satisfies many of the commonly desired theorems, but it is unknown to me for example if it satisfies a nice change of variables formula.

This is reffering to question 1.

There's a choice we have to make when defining a real-valued two-dimensional (or n dimensional) nonabsolute integral: would we rather have a class of integrable functions which include all divergences of differentiable functions, or would we want to have some sort of Fubini's Theorem working? This conflict was know from the early development of the HK integral and was also pointed out by Pfeffer in the book mentioned in Gerald Edgar's answer.

If we define the HK integral in the obvious way (call it the standard two dimensional HK integral), we get Fubini's theorem, but no fully general divergence theorem. Many authors made modifications on the definition of this integral, and an relatively satisfactory definition was given by Jarník, Kurzweil and Schwabik in "On Mawhin's approach to multiple nonabsolutely convergent integral", Casopis. Pest. Mat. There they defined the $M_1$-integral, which satisfies a fully general divergence theorem, and has a simple enough definition so we can prove some convergence theorems. It is shown though that this integral does not satisfy Fubini's theorem when the corresponding one-dimensional integrals are considered to be the HK integral. The original example from that paper can be used to show that, in a more general setting, an interval-based two-dimensional integral that satisfies a full divergence theorem will fail in some sense Fubini's theorem (see Proposição 2.5 here if you are not afraid of reading in portuguese).

Other problem that is frequently overlooked when defining interval-based two-dimensional nonabsolute integrals is that the integral can be sensitive to rotations, that is, we can get integrable functions such that a certain rotation of that function is not integrable. We have this unpleasant effect for the $M_1$-integral and even for the standard two-dimensional HK integral (see main theorem of "K teorii vícerozmerného integrálu", Casopis. Pest. Mat. by K. Kartak, if you are not afraid of reading czech, or Proposição 1.8 in the aforementioned Thesis, which is for the $M_1$-integral but easily adaptable to the standard two-dimensional HK one).

Then there is a new challenge: trying to define an integral which is not based on intervals but that still will be simple enough to prove convergence theorems. Kurzweil himself defined an integral where the domain is partitioned into sets with boundaries continuously differentiable by parts; it's a lot of trouble even to prove Saks' Lemma for this integral. See also this article for an integral where we use triangular partitions. This integral satisfies many of the commonly desired theorems, but it is unknown to me for example if it satisfies a nice change of variables formula.

This is referring to question 1.

There's a choice we have to make when defining a real-valued two-dimensional (or n dimensional) nonabsolute integral: would we rather have a class of integrable functions which include all divergences of differentiable functions, or would we want to have some sort of Fubini's Theorem working? This conflict was know from the early development of the HK integral and was also pointed out by Pfeffer in the book mentioned in Gerald Edgar's answer.

If we define the HK integral in the obvious way (call it the standard two dimensional HK integral), we get Fubini's theorem, but no fully general divergence theorem. Many authors made modifications on the definition of this integral, and an relatively satisfactory definition was given by Jarník, Kurzweil and Schwabik in "On Mawhin's approach to multiple nonabsolutely convergent integral", Casopis. Pest. Mat. There they defined the $M_1$-integral, which satisfies a fully general divergence theorem, and has a simple enough definition so we can prove some convergence theorems. It is shown though that this integral does not satisfy Fubini's theorem when the corresponding one-dimensional integrals are considered to be the HK integral. The original example from that paper can be used to show that, in a more general setting, an interval-based two-dimensional integral that satisfies a full divergence theorem will fail in some sense Fubini's theorem (see Proposição 2.5 here if you are not afraid of reading in portuguese).

Other problem that is frequently overlooked when defining interval-based two-dimensional nonabsolute integrals is that the integral can be sensitive to rotations, that is, we can get integrable functions such that a certain rotation of that function is not integrable. We have this unpleasant effect for the $M_1$-integral and even for the standard two-dimensional HK integral (see main theorem of "K teorii vícerozmerného integrálu", Casopis. Pest. Mat. by K. Kartak, if you are not afraid of reading czech, or Proposição 1.7 in the aforementioned Thesis, which is for the $M_1$-integral but easily adaptable to the standard two-dimensional HK one).

Then there is a new challenge: trying to define an integral which is not based on intervals but that still will be simple enough to prove convergence theorems. Kurzweil himself defined an integral where the domain is partitioned into sets with boundaries continuously differentiable by parts; it's a lot of trouble even to prove Saks' Lemma for this integral. See also this article for an integral where we use triangular partitions. This integral satisfies many of the commonly desired theorems, but it is unknown to me for example if it satisfies a nice change of variables formula.

This is reffering to question 1.

There's a choice we have to make when defining a real-valued two-dimensional (or n dimensional) nonabsolute integral: would we rather have a class of integrable functions which include all divergences of differentiable functions, or would we want to have some sort of Fubini's Theorem working? This conflict was know from the early development of the HK integral and was also pointed out by Pfeffer in the book mentioned above.

If we define the HK integral in the obvious way (call it the standard two dimensional HK integral), we get Fubini's theorem, but no fully general divergence theorem. Many authors made modifications on the definition of this integral, and an relatively satisfactory definition was given by Jarník, Kurzweil and Schwabik in "On Mawhin's approach to multiple nonabsolutely convergent integral", Casopis. Pest. Mat. There they defined the $M_1$-integral, which satisfies a fully general divergence theorem, and has a simple enough definition so we can prove some convergence theorems. It is shown though that this integral does not satisfy Fubini's theorem when the corresponding one-dimensional integrals are considered to be the HK integral. The original example from that paper can be used to show that, in a more general setting, an interval-based two-dimensional integral that satisfies a full divergence theorem will fail in some sense Fubini's theorem (see Proposição 2.5 here if you are not afraid of reading in portuguese).

Other problem that is frequently overlooked when defining interval-based two-dimensional nonabsolute integrals is that the integral can be sensitive to rotations, that is, we can get integrable functions such that a certain rotation of that function is not integrable. We have this unpleasant effect for the $M_1$-integral and even for the standard two-dimensional HK integral (see main theorem of "K teorii vícerozmerného integrálu", Casopis. Pest. Mat. by K. Kartak, if you are not afraid of reading czech, or Proposição 1.8 in the aforementioned Thesis, which is for the $M_1$-integral but easily adaptable to the standard two-dimensional HK one).

Then there is a new challenge: trying to define an integral which is not based on intervals but that still will be simple enough to prove convergence theorems. Kurzweil himself defined an integral where the domain is partitioned into sets with boundaries continuously differentiable by parts; it's a lot of trouble even to prove Saks' Lemma for this integral. See also this article for an integral where we use triangular partitions. This integral satisfies many of the commonly desired theorems, but it is unknown to me for example if it satisfies a nice change of variables formula.

This is reffering to question 1.

There's a choice we have to make when defining a real-valued two-dimensional (or n dimensional) nonabsolute integral: would we rather have a class of integrable functions which include all divergences of differentiable functions, or would we want to have some sort of Fubini's Theorem working? This conflict was know from the early development of the HK integral and was also pointed out by Pfeffer in the book mentioned in Gerald Edgar's answer.

If we define the HK integral in the obvious way (call it the standard two dimensional HK integral), we get Fubini's theorem, but no fully general divergence theorem. Many authors made modifications on the definition of this integral, and an relatively satisfactory definition was given by Jarník, Kurzweil and Schwabik in "On Mawhin's approach to multiple nonabsolutely convergent integral", Casopis. Pest. Mat. There they defined the $M_1$-integral, which satisfies a fully general divergence theorem, and has a simple enough definition so we can prove some convergence theorems. It is shown though that this integral does not satisfy Fubini's theorem when the corresponding one-dimensional integrals are considered to be the HK integral. The original example from that paper can be used to show that, in a more general setting, an interval-based two-dimensional integral that satisfies a full divergence theorem will fail in some sense Fubini's theorem (see Proposição 2.5 here if you are not afraid of reading in portuguese).

Other problem that is frequently overlooked when defining interval-based two-dimensional nonabsolute integrals is that the integral can be sensitive to rotations, that is, we can get integrable functions such that a certain rotation of that function is not integrable. We have this unpleasant effect for the $M_1$-integral and even for the standard two-dimensional HK integral (see main theorem of "K teorii vícerozmerného integrálu", Casopis. Pest. Mat. by K. Kartak, if you are not afraid of reading czech, or Proposição 1.8 in the aforementioned Thesis, which is for the $M_1$-integral but easily adaptable to the standard two-dimensional HK one).

Then there is a new challenge: trying to define an integral which is not based on intervals but that still will be simple enough to prove convergence theorems. Kurzweil himself defined an integral where the domain is partitioned into sets with boundaries continuously differentiable by parts; it's a lot of trouble even to prove Saks' Lemma for this integral. See also this article for an integral where we use triangular partitions. This integral satisfies many of the commonly desired theorems, but it is unknown to me for example if it satisfies a nice change of variables formula.