As is well known, the Fubini theorem is very easy to be used in $\mathbb{R}^n$ due to the global coordinate of $\mathbb{R}^n$, because $dV_{s+t}=dV_{s}\times dV_{t}$ ($dV_{t}$ denotes the t-dimensional lebesgur measure).
However, in a Riemannian manifold $(X,\omega)$, we have no global coordinate.
Question: How can we split the volume form $dV_{\omega}$ into the product of 2 measures? Can the partition of unity work? If it can work, can anyone give some details?