Riemannian volume of a compactly supported smooth density

Let $(M,g)$ be a Riemannian manifold and assume countable choice. For a smooth compactly supported function $f:M\to \mathbb{R}$, define ${\int }_{M}f{\mu }_g$ by the intrinsic smooth density integral. More generally every compactly supported signed smooth density $\sigma$ on $M$ has its existing intrinsic integral, independently of a Riemannian metric.

The riemannian volume density is coordinate independent makes $f{\mu }_g$ a smooth compactly supported density. Apply Integral of a compactly supported smooth density and Orientation-free density integration and its properties; The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) is inherited precisely at chart-partition selection. In a chart the summand is the integral of the partition-weighted coefficient $f\sqrt{\det G}$. On a zero-manifold it is the finite sum of scalar density values, and empty support gives zero. No orientation is required.

Lee, Propositions 15.29–15.33 and Corollary 15.34, pp.389–391; density construction and integration pp.428–433.