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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.

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You can extend the definition by asserting that given the doubling property above, $\exists C, \kappa \geq 0$ such that $\forall x \in M, r \geq 0$ and $\theta \geq 1$, $B(x, \theta r) \leq C \theta^{\kappa} B(x, r)$.

This exactly shows that given a manifold $M$ and its associated geodesic distance and measure; is a space of homogeneous type. The spaces of Coifmann and Weiss that extend the use of Hardy spaces would contain these type. See Extensions of Hardy Spaces and their use in Analysis.

Further to this, if the manifold $M$ has non-negative Ricci curvature then the above doubling space condition holds. For a comprehensive review of this particular property see Lott and Villani Weak Curvature Conditions and Functional Inequalities.

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