I am looking for references on integration on non-compact Riemannian manifolds, specially on the change of variables theorem.
In particular I have non-compact manifold $M$ and I have an integral (in most interesting cases for me improper integral)
$$\int_M f(x) \mu(dx)$$
Where $\mu$ is standard volume measure on $M$ generated by the metric tensor $g$ on $M$. And I need to make change of variables $y=\phi (x)$, where $\phi$ is a diffeomorphism $\phi: M \to M$. Function $f$ may be unbounded, manifold $M$ may be unbounded (say $M= \mathbb{S}^1 \times \mathbb{R}^2$).
I have found reference book only for compact manifolds and bounded functions.
Please help with reference applicable to my case. Thank you very much!