Normed versus topological groups: Dichotomy and duality

Normed groups 1019. The Semigroup TheoremThis section, just as the preceding one, is focussed on metrizable locally compact topologicalgroups. Since a locally compact normed group possesses an invariant Haar-measure,much of the theory developed there and here goes over to locally compact normed groups– for details see [Ost-LB3]. In this section G is again a normed locally compact topologicalgroup. The aim here is to prove a generalization to the normed group setting of thefollowing classical result due to Hille and Phillips [H-P, Th. 7.3.2] (cf. Beck et al. [BCS,Th. 2], [Be]) in the measurable case, and to Bingham and Goldie [BG] in the Baire case;see [BGT, Cor. 1.1.5].Theorem 9.1 (Category (Measure) Semigroup Theorem). For an additive Baire (resp.measurable) subsemigroup S of R + , the following are equivalent:(i) S contains an interval,(ii) S ⊇ (s, ∞), for some s,(iii) S is non-meagre (resp. non-null).We will need a strengthening of the Kestelman-Borwein-Ditor Theorem, Th. 1.1.involving two sets. First we capture a key similarity (their topological ‘common basis’,adapting a term from logic) between the Baire and measure cases. Recall ([Rog2, p. 460])the usage in logic, whereby a set B is a basis for a class C of sets whenever any memberof C contains a point in B.Theorem 9.2 (Common Basis Theorem). For V, W Baire non-meagre in a group Gequipped with either the norm or the density topology, there is a ∈ G such that V ∩ (aW )contains a non-empty open set modulo meagre sets common to both, up to translation.In fact, in both cases, up to translation, the two sets share a norm G δ subset which isnon-meagre in the norm case and non-null in the density case.Proof. In the norm topology case if V, W are Baire non-meagre, we may suppose thatV = I\M 0 ∪ N 0 and W = J\M 1 ∪ N 1 , where I, J are open sets. Take V 0 = I\M 0 andW 0 = J\M 1 . If v and w are points of V 0 and W 0 , put a := vw −1 . Thus v ∈ I ∩ (aJ). SoI ∩ (aJ) differs from V ∩ (aW ) by a meagre set. Since M 0 ∪ N 0 may be expanded to ameagre F σ set M, we deduce that I\M and J\M are non-meagre G δ -sets.In the density topology case, if V, W are measurable non-null let V 0 and W 0 be the setsof density points of V and W. If v and w are points of V 0 and W 0 , put a := vw −1 . Thenv ∈ T := V 0 ∩ (aW 0 ) and so T is non-null and v is a density point of T. Hence if T 0comprises the density points of T, then T \T 0 is null, and so T 0 differs from V ∩ (aW ) bya null set. Evidently T 0 contains a non-null closed, hence G δ -subset (as T 0 is measurablenon-null, by regularity of Haar measure).