I trying to figure out why this is result is true :
If $F$ is a $C^1$-foliation with $C^1$-leaves, then it is absolutely continuous, where absolutely continuity here means that given a foliation chart $(U,h)$ and $L$ some transversal, there exists a measurable family of positive measurable functions $j_x : F_U(x) \to \mathbb{R}$ such that for any measurable set $A \subset U$, we have that $m(A) = \int_L \int_{F_U(x)} \mathbb{1}_A(x,y)j_x(y)dm_{F(x)}(y)dm_L(x)$ where $m$ is our volume.
I've read that this is a direct application of Fubini Theorem, but since I don't know much about foliation and differential geometry, I don't really feel confident about this...
Any help is very greatly appreciated.
EDIT : this question has been put on hold because it lacks some context. So here it is. I'm interesting in proving that ergodicity of $C^2$ Anosov on compact riemannian manifold. I'm following Brin & Stuck Introduction to dynamical systems. The key idea of the proof is to prove that the stable/unstable foliation is absolutely continuous, as defined above. At one point, authors show that absolutely continuous implies transversely absolute continuity. To do that, they introduce some auxiliary foliation that is $C^1$, while explaining that it is obviously absolutely continuous and transversely absolutely continuous. I've read this in differents books/ressources, as a direct application of Fubini theorem. The context is pretty standard in DS, so maybe I should be posting this question of MO...
EDIT 2 : Some more context here (especially the line "...basic results from calculus show that a version of Fubini’s theorem holds..." This is precisely the claim for which I'm looking a proof. Since I'm working only on the torus, it might goes like this :
Let $F$ a $C^1$ foliation with $C^1$-leaves on the torus $\mathbb{T}^2$, with $\lambda$ the Lebesgue measure. Let $U$ a foliation chart. Then, for any $E \subset U \subset \mathbb{T}^2$ measurable, we have : $\lambda(E) = \int_L \int_{F_U(x)} \mathbb{1}_E(x,y)\delta_x(y) dm_{F(x)(y)} dm_L(x)$ with $L$ any local transversal and $(\delta_x)_x$ a family of positive measurable functions.