Bounded common extensions of charges

Abstract:

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