Nowhere densely generated properties in topological measure theory

Abstract

A topologicalproperty £Pis said to be nowhere densely generated in a class C of topological spaces if each Xg C has £P whenever every nowhere dense closed subset of X has £p[5]. For example, Katetov showed in [4] that the subspace of nonisolated points of a Tx -space X is compact if each nowhere dense closed subset is compact. This means that compactness is a nowhere densely generated property in the classof 7\-spaces without isolated points. More generally, it was showed in [1] [5] that [k, ^-compactness is also a nowhere densely generated property in the same class. Other nowhere densely generated properties were investigated in [1], a-closed-completeness, a-compactness and pseudo-(≪, ^-compactness. The purpose of this paper is to consider nowhere densely generated properties in topologicalmeasure theory. In this paper we examine measurecompactness, (weak) Borel measure-completeness, Borel measure-compactness, preRadon-ness and Radon-ness. Terminologies and notations are due to [3]. we denote by <B(X) (<B*(X)) the Borel (Baire) a-algebra in a space X. A Borel (Baire) measure fiis a <7-additive non-negative real-valued set function on £B(X){<B*{X)). We assume that all measures are finite(i.e.ft(X)<oo). A measure ft which is fi{X)―\is called a probability. A Borel measure ft is called regular (Radon) if for each Bz<B{X) ft(B) is the supremum of measures of closed (compact) subsets contained to B. A nonempty family JL of sets is called directed upwards if for each A, BeJL there exists CgJI such that AuBczC. A Borel measure ftis called weakly r-additive if for each directed upwards open cover Jl of X, ft(X)=sup{ft(U): UzJl). A Borel measure ft is called r-additive if for given open subset V and a directed upwards open cover Jl of V, ft(V)=s\ip{ft(U):UzJl). Regularity and r-additivity of Baire measures are also defined by the same way. We assume all spaces are T2.