Discrete approximation of spaces of homogeneous type

Abstract

In this note we combine the dyadic families introduced by M. Christ in (Colloq. Math. 60/61(2):601-628, 1990) and the discrete partitions introduced by J.M. Wu in (Proc. Am. Math. Soc. 126(5):1453-1459, 1998) to get approximation of a compact space of homogeneous type by a uniform sequence of finite spaces of homogeneous type. The convergence holds in the sense of a metric built on the Hausdorff distance between compact sets and on the Kantorovich-Rubinshtein metric between measures.