Logics for contact and measure

Abstract We enrich contact algebras with a new binary relation that compares the size of regions, and provide axiom systems for various logics of contact and measure. Our contribution is three-fold: (i) we characterize the relations on a Boolean algebra that derive from an unbounded measure, thereby generalizing a famous result of Kraft, Pratt and Seidenberg; (ii) for all $n \geq 1$, we axiomatize the logic of regular closed sets of $\mathbb R^{n}$ with null boundary; (iii) considering a broad class of equational theories that contains all logics of contact, we prove that they all have unary or finitary unification, and that unification and admissibility are decidable.