Brin & Stuck defined in Introduction to dynamical system two notions:

  • That of a absolutely continuous foliation : given any foliated chart $U$ on some Riemannian manifold $M$ (with foliation $W$), and for any transversal $L$, there exists a family of positive measurable functions $(\delta_x)_x$ such that $m(E) = \int_L \int_{W(x)} \mathbb{1}_{E}(x,y) \delta_x(y) dm_{W(x)}(y) dm_L(y)$ with $E \subset U$ measurable and $m$ the Riemannian metric on $M$.

  • That of transversely absolutely continuous foliation : given two transversals $L_1$ and $L_2$, and $p : L_1 \to L_2$ the holonomy map, there is a positive measurable function $q : L_1 \to \mathbb{R}$ such that for any measurable set $A \subset L_1$, we have that $m_{L_2}(p(A)) = \int_{L_1}\mathbb{1}_A(z) q(z) dm_{L_1}(z)$.

In the proof of proposition 6.2.2., (whose object is to show that transversely absolutely continuous foliations are absolutely continuous) authors claim that $C^1$ foliations (with $C^1$-leaves) are automatically absolutely continuous and transversely continuous. I was looking for a proof of this claim, but in most references (Katok & Hasselblatt, Barreira & Pesin...) I found, authors just say that this is a direct application of some sort of Fubini theorem on the foliation, without explicitly giving a proof. Here, author gives a particular case when looking at foliations on $[0,1] \times [0,1]$. Since my knowledge on foliation and more generally in differential geometry is quite restricted, I'll be very happy is somebody could give me a proof/reference of this special (simpler) claim.

Thanks a lot!

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