Measure Theory in Convergence Spaces Part I: Sion-Carathéodory Extension in Net Convergence Groups

Abstract

We establish a Carathéodory-type extension theorem for premeasures with values in general separated abelian net convergence groups. Starting from hereditary exhaustivity on the source ring and intrinsic conditions for combining null errors, we construct a countably additive extension on a regulated Boolean sigma-algebra, with full upper and lower regularity in the original convergence. Completeness is required only on the limit-stable hull of the source range. Under closed summation for arbitrary directed rows and finite-response reflection, we identify the regulated domain with the splitting class of Sion's inner and outer processes. For a Dedekind-complete real vector lattice, we show that sequential closed summation on a bounded interval is equivalent to weak sigma-distributivity of the generated band, giving a converge-theoretic interpretation of the concept. Source representations allow us to recover the partially ordered-group extension of Volauf and the finite-variation extension of Kusraev and Malyugin from the general theorem. We then treat finite semivariation by adjoining dominated coordinate variations to an additive lift of the premeasure. The resulting extension is defined on the full semivariation regulator domain and preserves semivariation exactly, without requiring decomposability or the countable sup property. It includes measures of infinite vector variation.