Normed versus topological groups: Dichotomy and duality

100 N. H. Bingham and A. J. OstaszewskiCorollary 8.2 (Piccard Theorem, [Pic1], [Pic2]). For S Baire and non-meagre in thenorm topology, the difference sets SS −1 and S −1 S have e as interior point.First Proof. Apply the preceding Theorem, since by the First Verification Theorem (Th.6.2), the condition (wcc) holds. Second Proof. Suppose otherwise. Then, as before, for each positive integer n we mayselect z n ∈ B 1/n (e)\(S −1 S). Since z n → e, by the Kestelman-Borwein-Ditor Theorem(Cor. 6.4), for quasi all s ∈ S there is an infinite M s such that {sz m : m ∈ M s } ⊆ S.Then for any m ∈ M s , sz m ∈ S , i.e. z m ∈ SS −1 , a contradiction. Corollary 8.3 (Steinhaus Theorem, [St], [We]; cf. Comfort [Com, Th. 4.6 p. 1175] ,Beck et al. [BCS]). In a normed locally compact group, for S of positive measure, thedifference sets S −1 S and SS −1 have e as interior point.Proof. Arguing as in the first proof above, by the Second Verification Theorem (Th. 7.5),the condition (wcc) holds and S, in the density topology, is Baire and non-meagre (by theCategory-Measure Theorem, Th. 7.2). The measure-theoretic form of the second proofabove also applies.The following corollary to the Steinhaus Theorem Th. 6.10 (and its Baire categoryversion) have important consequences in the Euclidean case. We will say that the groupG is (weakly) Archimedean if for each r > 0 and each g ∈ G there is n = n(g) such thatg ∈ B n where B := {x : ‖x‖ < r} is the r-ball.Theorem 8.4 (Category (Measure) Subgroup Theorem). For a Baire (resp. measurable)subgroup S of a weakly Archimedean locally compact group G, the following are equivalent:(i) S = G,(ii) S is Baire non-meagre (resp. measurable non-null).Proof. By Th. 8.1, for some r-ball B,and hence G = ⋃ n Bn = S.B ⊆ SS −1 ⊆ S,We will see in the next section a generalization of the Pettis extension of Piccard’sresult asserting that, for S, T Baire non-meagre, the product ST contains interior points.As our approach will continue to be bitopological, we will deduce also the Steinhaus resultthat, for S, T non-null and measurable, ST contains interior points.