  We show that in doubling, geodesic metric measure spaces (including, for
example, Euclidean space), sets of positive measure have a certain large-scale
metric density property. As an application, we prove that a set of positive
measure in the unit cube of $\mathbb{R}^d$ can be decomposed into a controlled
number of subsets that are "well-connected" within the original set, along with
a "garbage set" of arbitrarily small measure. Our results are quantitative,
i.e., they provide bounds independent of the particular set under
consideration.
