Quantitative metric density and connectivity for sets of positive measure

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.