Variants of the change-of-variables formula

Question

Consider the following change of variables formula for $f:X\rightarrow Y$, that holds for any "reasonable" $g:B\subseteq Y \rightarrow \mathbb{R}$ and $A\subseteq X$

$$ \int_B g(x)\ {\rm d}(x)=\int_Ag\bigl(f(u)\bigr)\>|J_f(u)|\>{\rm d}(u),$$where $|J_f(u)|$ denotes the jacobian, and $f(A)=B$ and $f$ is essentially injective.

I have two questions regarding this formula:

1) What would be the most general $g$ that would pass as "reasonable" and the most general notion of integral for that the transformation formula holds ?

2) One can define that a function $f:X\rightarrow Y$ is area-preserving, if $$\mu\bigl(f^{-1}(B)\bigr)=\mu(B)\qquad\forall B\subset Y\ \tag{1}$$and then show, if $f$ is essentially injective, as it is done in this answer (to which this question is a follow-up), by using the above formula and the fact that $(1)$ is for such an $f$ equivalent to $\mu(f(A))=\mu(A)\ \forall A\subseteq X$, that $(1)$ is equivalent to $\det( J_f(u))=\pm 1$ for all $u\in A$ and all $A\subseteq X$.
Now if moreover we had $J_f(u)=+1$, then intuitively (by analogy to linear algebra) this should mean that $f$ is also orientation-preserving. But is there is geometric definition of "orientation-preserving" other than defining by $\det (J_f(u)) =+1$ (just as there is a geometric definition of "area-preserving", different than defining "area-preserving to mean $\det( J_f(u)) =±1$), so that I could you that definition and then try to show, maybe by using the change of variables formula similar to the linke answer, that that implies $\det( J_f(u))=+1$ ?

(I'd also be thankful for a reference to books or articles, if these answer my question, to spare you from typing a lot - although I don't have anything against a complety written answer :) )

Answer 1

As for your first question, this is an old and extensively studied problem. This depends on the kind of integral used. For example, the Kurzweil-Henstock integral has a nice multidimensional change of variable theorem (see any serious book on the Kurzweil-Henstock integral). There is also the so-called geometric integral, with nice Stokes formulae. Notice also that it has been shown that Fubini's theorem and the change of variable theorem are in some sense contradictory each with the other: you cannot build an integral that has at the same time a "good" Fubini theorem and a "good" change of variable theorem (I think this is shown in a book of Pfeffer).

Regarding your question about orientation, I think the right question is first of all: what is orientation? the only simple way to define it seems to be, indeed, with the notion of determinant. So, I don't think there is something "bad" by defining $f$ to be "orientation preserving" using the determinant of the Jacobian. Furthermore, it seems necessary to use the differentials of $f$ because your notion of "orientation preserving" is essential local (have you another "global" interpretation?). On the contrary, the fact that |det(J)|=1 implies area preservation is a theorem, that depends on some hypothese about $f$ (at least differentiability). So, the two things are not similar. And to answer more prosaically, I have personally never seen such a notion.