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 T μ which is some kind of abstract density topology and, simultaneously, a natural generalization of the classical density topology T d . We focus on separation axioms of topologies T μ . We show that considered topologies are regular and not normal. In some cases we prove that T μ gives a Tychonoff space. We also consider homeomorphisms between topologies T μ . It is shown that for any atomless measure ν the topology T ν is homeomorphic to T d . However, there are measures μ generating topologies quite different than T d , for example separable and not connected. Some cases deliver us examples of spaces which are not Lindelof but they are separable.