On weak null-additivity of monotone measures

Abstract

In this note, the relations between weak null-additivity and pseudometric generating property of monotone measures are discussed. We show that on finite continuous monotone measure spaces (X, F, μ), if measurable space (X, F) is S-compact (especially, if X is countable), then the weak null-additivity is equivalent to pseudometric generating property. We put a question: abandoning the S-compactness condition, does the equivalence remain valid? © 2012 Springer-Verlag.