A measure can be defined, by the Riesz representation theorem, by giving a positive linear functional Λ on the space C 0 (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 g such that for any coordinate chart (U, φ), Λ Λ f = ∫ ∫ U f d μ μ g = ∫ ∫ φ φ ( U ) f ∘ ∘ φ φ − − 1 ( x ) | det g | d x 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 0 (M) by means of a partition of unity.