I am learning the Geometric Measure Theory, and curious about how to generalize the covering lemma on manifold. However I am stuck at the doubling property:
Let $(X,d,\mu)$ be a metric space with measure. If there exists a constant $C$, satisfies $$\mu(B(x,2r))\leq C\mu(B(x,r)),$$ then we call $X$ satisfies doubling property.
Which Riemannian manifold satisfies doubling space? Is there any condition on manifold?
Any advice is helpful. Thank you.