Extension of kappa-additive measures

Abstract

We establish an extension theorem for kappa-additive probability measures, where kappa is an arbitrary uncountable cardinal. Starting from a <kappa-algebra, we define an outer measure using coverings of cardinality at most kappa and characterise measurable sets by approximation in outer measure of symmetric differences. We prove that these sets form a kappa-algebra carrying a complete kappa-additive extension of the original measure. The extension is unique on the generated kappa-algebra, and its completion is the minimal domain admitting a complete kappa-additive extension. We also establish attainment of the covering infimum and show why the closure assumption on the initial algebra is essential. Examples include a strict extension of classical Lebesgue measure under the additional hypothesis add N = aleph_1 < cov N.