Bounded common extensions of charges

Abstract

Let A \mathcal {A} and B \mathcal {B} be fields of subsets of a set X and let μ : A → R \mu :\mathcal {A} \to {\mathbf {R}} and ν : B → R \nu :\mathcal {B} \to {\mathbf {R}} be consistent, bounded, finitely additive measures (i.e., charges). We give necessary and sufficient conditions for μ \mu and ν \nu to have a bounded common extension to A ∨ B \mathcal {A} \vee \mathcal {B} . Conditions on A \mathcal {A} and B \mathcal {B} are given under which any bounded consistent charges μ \mu and ν \nu have a bounded common extension.