I'm having some hard times understanding a step in the proof of the volume rigidity theorem for hyperbolic compact 3-manifolds in this paper: https://arxiv.org/pdf/math/9802022.pdf .
I have the set $X=S^{2}_{\infty}\ x \ S^{2}_{\infty} \ x S^{2}_{\infty}$, where $S^{2}_{\infty}$ is the Riemann sphere. I know that the space $T$ of regular oriented ideal tetrahedra is a full measure subset of X. Moreover, I also know that the subset of $T^{G}$ of $T$ made up of those tetrahedron which are mapped to $T$ by a smooth equivariant measurable map $\bar{f}:S^{2}_{\infty}\rightarrow S^{2}_{\infty}$ is of full measure.
Now, the part I'm struggling to understand is the following: by Fubini's theorem there is a $v_{0}\in S^{2}_{\infty}$ such that almost all tetrahedra in $T$ with first vertex $v_{0}$ are in $T^{G}$.
How does this occur?