linguistic:paraphrasing
Canonical measure and volume form

In analogy with the case of surfaces, a metric tensor on an n -dimensional paracompact manifold M gives rise to a natural way to measure the n -dimensional volume of subsets of the manifold. The resulting natural positive Borel measure allows one to develop a theory of integrating functions on the manifold by means of the associated Lebesgue integral.

A measure can be defined, by the Riesz representation theorem, by giving a positive linear functional Λ on the space C (M) of compactly supported continuous functions on M. More precisely, if M is a manifold with a (pseudo-)Riemannian metric tensor g, then there is a unique positive Borel measure μ such that for any coordinate chart (U, φ), Λ f = ∫ U f d μ g = ∫ φ (U) f ∘ φ − 1 (x) | det g | d x {\displaystyle \Lambda f=\int _{U}f\,d\mu _{g}=\int _{\varphi (U)}f\circ \varphi ^{-1}(x){\sqrt {\left|\det g\right|}}\,dx} for all f supported in U. Here det g is the determinant of the matrix formed by the components of the metric tensor in the coordinate chart. That Λ is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of variables. It extends to a unique positive linear functional on C (M) by means of a partition of unity.

If M is also oriented, then it is possible to define a natural volume form from the metric tensor. In a positively oriented coordinate system (x,..., x) the volume form is represented as ω = | det g | d x 1 ∧ ⋯ ∧ d x n {\displaystyle \omega ={\sqrt {\left|\det g\right|}}\,dx^{1}\wedge \cdots \wedge dx^{n}} where the dx are the coordinate differentials and ∧ denotes the exterior product in the algebra of differential forms. The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure.