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The regular open subsets of a topological space form a Boolean algebra, wherethe `join' of two regular open sets is the interior of the closure of theirunion. A `credence' is a finitely additive probability measure on this Booleanalgebra, or on one of its subalgebras. We develop a theory of integration forsuch credences. We then explain the relationship between credences, residualcharges, and Borel probability measures. We show that a credence can berepresented by a normal Borel measure, augmented with a `liminal structure',which specifies how two or more regular open sets share the probability mass oftheir common boundary. In particular, a credence on a locally compact Hausdorffspace can be represented by a normal Borel measure and a liminal structure onthe Stone-\v{C}ech compactification of that space. We also show how credencescan be represented by Borel measures on the Stone space of the underlyingBoolean algebra of regular open sets. Finally, we show that these constructionsare functorial.

Measure and integration on Boolean algebras of regular open subsets in a topological space