Compactness in spaces of inner regular measures and a general Portmanteau lemma

Abstract

This paper may be understood as a continuation of Topsoe's seminal paper [F. Topsoe, Compactness in spaces of measures, Studia Math. 36 (1970) 195-212] to characterize, within an abstract setting, compact subsets of finite inner regular measures w.r.t. the weak topology. The new aspect is that neither assumptions on compactness of the inner approximating lattices nor nonsequential continuity properties for the measures will be imposed. As a providing step also a generalization of the classical Portmanteau lemma will be established. The obtained characterizations of compact subsets w.r.t. the weak topology encompass several known ones from literature. The investigations rely basically on the inner extension theory for measures which has been systemized recently by König [H. König, Measure and Integration, Springer, Berlin, 1997; H. König, On the inner DaniellStone and Riesz representation theorems, Doc. Mat. 5 (2000) 301-315; H. König, Measure and integration: An attempt at unified systematization, Rend. Istit. Mat. Univ. Trieste 34 (2002) 155-214]. IntroductionThe most influential result concerning compactness of spaces of measures has been presented by Prokhorov [17] for probability Borel-measures on Polish spaces. His equivalent characterization of relative compactness by uniform tightness has turned out to be an important tool to check convergence in law for many stochastic processes. Therefore nowadays this criterion may be found in most of standard textbooks of probability theory (see e.g. [3,4,7,16]). Another characterization of compact sets of Borel probability measures on Polish spaces has been shown by Huber and Strassen [8] in terms of a continuity property that the upper envelopes of these sets satisfy. Prokhorov as well as Huber and Strassen used the so called topology of weak convergence which is derived from the weak * topology on the algebraic dual of the space of bounded continuous mappings. Of course this topology may be extended to spaces of finite Baire-measures on general topological spaces. This has been done by Varadarajan [22] who has also found an equivalent characterization for compact sets. However the topology of weak convergence relies on hidden regularity properties. Finite Baire-measures are inner regular w.r.t. the functionally closed sets, and in the special context of metrizable spaces they coincide with the finite Borel-measures, being inner regular w.r.t. the closed subsets. But in general finite Borel-measures are not inner regular w.r.t. the closed subsets. So for spaces of such measures the topology of weak convergence is not a reasonable concept because the measures are not uniquely determined by the restrictions of the integrals to bounded continuous mappings. Furthermore it seems to be necessary to impose regularity for the measures to find tractable extensions of the topology of weak convergence. ✩