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

This paper may be understood as a continuation of Topsoeʹs 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 Daniell-Stone 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].