Boolean algebras: Convergence and measure

We study topological and categorical aspects of the extension of σ-additive measures from a field of sets to the generated σ-field within a category of Boolean algebras carrying initial sequential convergences with respect to 2-valued homomorphisms. We describe the relationship between σ-additivity and sequential continuity of finitely additive measures. A key role is played by the epireflective subcategory of absolutely sequentially closed objects. In case of fields of sets such objects are exactly σ-fields. The results provide information about basic notions of probability theory: events, probability measures, and random functions.