Normed versus topological groups: Dichotomy and duality

112 N. H. Bingham and A. J. Ostaszewskicontradicting the unboundedness of f(u n ).The Generalized Mehdi Theorem, together with the Averaging Lemma, implies theclassical result below and its generalizations.Theorem 10.14 (Császár-Ostrowski Theorem [Csa], [Kucz, p. 210]). A convex functionf : R →R bounded above on a set of positive measure (resp. non-meagre set) is continuous.Theorem 10.15 (Topological Császár-Ostrowski Theorem). A 1 2-convex function f :G → R on a normed topological group, bounded above on a non-meagre subset, is continuous.Appeal to the Generalized Borwein-Ditor Theorem yields the following result, whichrefers to Radon measures, for which see Fremlin [Fre-4].Theorem 10.16 (Haar-measure Császár-Ostrowski Theorem). A 1 2-convex function f :G → R on a normed topological group carrying a Radon measure, bounded above on a setof positive measure, is continuous.11. Automatic continuity: the Jones-Kominek TheoremThis section is dedicated to generalizations to normed groups and to a more general classof topological groups of the following result for the real line. Here we regard R as a vectorspace over Q and so we say that T is a spanning subset of R if any real number is a finiterational combination of members of T. See below for the definition of an analytic set.Theorem 11.1 (Theorems of Jones and Kominek). Let f be additive on R and eitherhave a continuous restriction, or a bounded restriction, f|T , where T is some analyticset spanning R. Then f is continuous.The result follows from the Expansion Lemma and Darboux’s Theorem (see below)that an additive function bounded on an interval is continuous. In fact the boundedcase above (Kominek’s Theorem, [Kom2]) implies the continuous case (Jones’s Theorem,[Jones1], [Jones2]), as was shown in [?]. [OC] develops limit theorems for sequencesof functionals whose properties are given on various kinds of spanning sets includingspanning in the sense of linear rational combinations. Before stating the current generalizationswe begin with some preliminaries on analytic subsets of a topological group. Werecall ([Jay-Rog, p.11], or [Kech, Ch. III] for the Polish space setting) that in a Hausdorffspace X a K-analytic set is a set A that is the image under a compact-valued, uppersemi-continuous map from N N ; if this mapping takes values that are singletons or empty,the set A is said to be analytic. In either case A is Lindelöf. (The topological notion of