Weak porosity on metric measure spaces

We characterize the subsets E of a metric space X with doubling measure whose distance function to some negative power dist(·,E)-α belongs to the Muckenhoupt A 1 class of weights in X . To this end, we introduce the weakly porous sets in this setting, and show that, along with certain doubling-type conditions for the sizes of the largest E -free holes, these sets characterize the mentioned A 1 -property. We exhibit examples showing the optimality of these conditions, and simplify them in the particular case where the underlying measure satisfies a qualitative annular decay property. In addition, we use some of these distance functions as a new and simple method to explicitly construct doubling weights in Rⁿ that do not belong to A_∞.