Here is the setup. I have a submersion $f:X \to T$, where $X$ is manifold and $T$ is a torus (you can chose a circle for the beginning if it is simpler). The manifold $X$ has a volume form $\alpha$ that is can be locally decomposed as $\alpha' \wedge f^* \alpha_S$ where $\alpha_S$ is the standard volume form of the torus (its Haar measure). I want to compute the volume of $X$.
To do this, I look at $T_k \subset T$ the subset of $k$-torsion points (or roots of unity if we take the simplified case of a circle) for all $k\geq 1$. I write $X_k=f^{-1}(T_k)$. Then it happens that $T_k$ has a finite volume (for the transverse volume form $\alpha'$) and that $vol(X_k)/k^{dim(T)}$ converges as $k$ goes to $\infty$.
Can I conclude with this fact that the volume of $X$ is finite? and that it is equal to $lim(vol(X_k))$?
PS: beware that I do not know a priori if $f^{-1}(t)$ has a finite volume for a general $t\in T$.
