I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.


Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ${y} \in \mathbb{R}^m$, the image $g(\mathbb{R}^n \times \{y\})$ is an $n$-dimensional $C^1$ manifold, and, similarly, for a fixed ${x}$, the image $g(\{x\} \times \mathbb{R}^m)$ is an $m$-dimensional $C^1$ manifold. Let $\mathcal{H}^n$ and $\mathcal{H}^m$ be, respectively, the Hausdorff measures on these with respect to the intrinsic metric on them induced from $\mathbb{R}^{n+m}$. As mentioned in the comments below, they will be different from fiber to fiber, and, for example, it is not true that all these measures are identifiable.


**Original Question:** I wonder if a "Fubini's theorem" can be formulated and proven using integrals on these manifolds directly.\*\* I do NOT wish to pullback to $V$ via $g$,


Edit: Initially I stated "I do not want to contaminate my integral with the Jacobian!" In light of comments below, it will be impossible to bring in some type of Jacobian(S) into picture. Now, it looks obvious: We must take into account how fibers close in or expand away from one another at different neighborhoods. So, now I reiterate my question allowing this:


**Edited Question:** Is there "a Fubini's theorem" that equates an integral over $U$ to the iterated integrals (of the function probably multiplied with some Jaobian of the map $g$) over these fibers -- against their intrinsic Hausdorff measures.\*\*


A cartoon of the sought-for identity will look like: for a continuous real-valued function $ \phi: U \to \mathbb{R}$, 
$$ \int\_U \phi \ d\mathcal{L}^{n+m}= \int\_{?} \left(\int\_{?} \phi(x,y) \cdot Jacobian \ quantities \ from \ g \ d\mathcal{H}^n(x)\right) \ d\mathcal{H}^m(y) \ .$$


**Note:** I seem to have figured out one such formula but will wait longer for possible alternatives or references to known ones, if any exists.


I have the answer here: [Fubini's Theorem on Arbitrary Foliations](https://mathoverflow.net/questions/350952/fubinis-theorem-on-arbitrary-foliations)



A Curved/Warped Version of Fubini's Theorem
I have the answer here: [Fubini's Theorem on Arbitrary Foliations](https://mathoverflow.net/questions/350952/fubinis-theorem-on-arbitrary-foliations)


$$\int\_U f = \int\_{U\_{\eta\_0}} \left(\int\_{U\_\xi} f(\xi,\eta) \frac{|\det DG\_{U\_\xi} (\xi,\eta)| \cdot |\det DG\_{U\_{\eta\_0}} (\xi,\eta\_0)|}{|\det DG(\xi,\eta)|} \ d\mathcal{H}^m(\eta)\right) \ d\mathcal{H}^n(\xi) \ .$$


