Type: Article
Publication Date: 1980-02-01
Citations: 71
DOI: https://doi.org/10.2140/pjm.1980.86.389
Let Sf be a sublattice of 2* containing 0,X and let MR{^f) be the collection of all bounded nonnegative finitely additive measures defined on J&(£f) the algebra generated by £f which are ^-regular in the sense that μ(E) = Bupμ(L), LczE, Le £f, E eJ/(^), It is shown here that if £f x τz£f Λ are sublattices of 2 X and ju6MB(^i), then μ extends to a veMR(^2).Several applications are given. O Introduction* In previous papers see [1-4, 18, 19] we were concerned with regular extensions of measures and their applications to several different areas of mathematics.Typically one was given a μeMRi^i) and conditions were given for when μ extended to a veMR(Jέf 2 ) where J^ci^ were sublattices of 2 Σ .Sufficient conditions for the countable additivity of v to follow from that of μ were also given.In this paper we show that for finitely additive measures, regular extensions always exist.This theorem represents a significant extension of our main theorem of [3] in that now no connection between ^ and Jΐf 2 is required except that Sf 1 c ,g\.This theorem has a great many applications and some of them are given both to measure extensions and the related concept of measure repleteness, a concept studied in many special cases in [5; 7-10, 14-17].