Given a pseudo-Riemannian manifold $M$, let say for example Minkowski spacetime in $s>1$ dimensions, often we talk about integration on it. Namely the connection/metric definition gives us a measure on which we integrate. However, to me it is not so clear how to compute integrals on pseudo-Riemannian manifolds in full generality. For example (and these are my main questions): how does the change of variables work? What can we integrate over such a manifold?

A clear and detailed reference on integration on pseudo-Riemannian manifolds is warmly welcome.

Thank you!