Nowhere densely generated properties in topological measure theory (Q1080222)

A topological property \({\mathcal P}\) is said to be nowhere densely generated in a class \({\mathcal C}\) of topological spaces if each \(X\in {\mathcal C}\) has \({\mathcal P}\) whenever every nowhere dense close subset of X has \({\mathcal P}\). Compactness is a nowhere densely generated property in the class of \(T_ 1\)-spaces without isolated points as shown by Katětov. We examine some nowhere densely generated properties in topological measure theory like measure-compactness, (weak) Borel measure-completeness, Borel measure-compactness, pre-Radon-ness and Radon-ness. For example it is shown that measure-compactness is a nowhere densely generated property in the class of normal spaces which the cardinality of each closed discrete subspace is not real-valued measurable.