Let $M$ be a smooth Riemannian manifold with riemannian measure $\mu$. I don't suppose that $M$ is complete. Can we find a finite or countable disjoint collection of open (or closed) and relatively compact geodesic balls$(B_n)_{n\in\mathbb{N}}$ such that :

$$\mu\left(M\setminus\bigsqcup_{n\in\mathbb{N}}B_n\right) = 0$$ ? We might assume that $M$ has a bounded curvature. The $M$ I'm interested in is the intersection of a (complete) submanifold of $\mathbb{R}^d$ with a ball. The compactness restriction ensures that each ball is a "real" ball, and hasn't been cropped by the "edge" of $M$.