Density topologies for strictly positive Borel measures

Abstract

The aim of this paper is to describe density topologies generated by Borel complete regular measures. For any such measure μ a generalization of the classical Lebesgue differentiation theorem is proved here. Using it, we define a topology Tμ which is some kind of abstract density topology and, simultaneously, a natural generalization of the classical density topology Td.We focus on separation axioms of topologies Tμ. We show that considered topologies are regular and not normal. In some cases we prove that Tμ gives a Tychonoff space. We also consider homeomorphisms between topologies Tμ. It is shown that for any atomless measure ν the topology Tν is homeomorphic to Td. However, there are measures μ generating topologies quite different than Td, for example separable and not connected. Some cases deliver us examples of spaces which are not Lindelöf but they are separable.