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 $\text{dist}(\cdot ,E{)}^{-\alpha }$ 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 ${\mathbb{R}}^n$ that do not belong to $A_{\mathrm{\infty }}\mathrm{.}$