# $C^1$-foliation are absolutely continuous

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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