How to define positive measure on a Riemannian manifold

Question

I need a formal definition that distinguishes sets of zero measure and sets that have strictly positive measure on a Riemannian manifold $(M,g)$. I don't want to introduce alot of concepts for this (such as the specific values of the measure of the sets), I would much prefer a simple definition.

I have modified a definition of zero measure to obtain:

A subset ⊂ has strictly positive measure if there exists at least one chart (,) of such that the set (∩) has strictly positive measure in $\mathbb{R}^n$ (w.r.t. lebesgue measure on $\mathbb{R}^n$).

which I found in a previous question: "measure zero" and "measurable function" on Riemannian manifolds

Is this correct and is it an acceptable definition of Riemannian measure?

Answer 1

This is fine. But rather than defining what it means to have positive measure, it's much more common to define what it means to have measure zero, using the negation of your definition:

A subset $U\subseteq M$ has measure zero if for every smooth chart $(V,y)$ of $M$, the set $y(V\cap U)$ has measure zero in $\mathbb R^n$.

This is the definition I use in Introduction to Smooth Manifolds, and it's also used in Munkres's Analysis on Manifolds, Jeffrey Lee's Manifolds and Differential Geometry, Guillemin & Pollack's Differential Topology, and probably other books as well.