Normed versus topological groups: Dichotomy and duality

Normed groups 87j ∈ ω} which enumerates, for given ε > 0, a dense subset of the ε ball about e), yieldingthe following.Theorem 6.9 (Strong Compactness Theorem – modulo shift, cf. [BOst-StOstr]). LetA be a strongly right-shift compact subset of a separable normed topological group G.Then A is compactly strongly shift-covered, i.e. for any norm-open cover U of A, andany neighbourhood of e X there is a finite subset V of U, and for each member of V atranslator in N such that the corresponding translates of V cover A.Next we turn to the Steinhaus theorem, which we will derive in Section 8 (Th. 8.3)more directly as a corollary of the Category Embedding Theorem. For completeness werecall in the proof below its connection with the Weil topology introduced in [We].Definitions ([Hal-M, Section 72, p. 257 and 273]). 1. A measurable group (X, S, m) isa σ-finite measure space with X a group and m a non-trivial measure such that both Sand m are left-invariant and the mapping x → (x, xy) is measurability preserving.2. A measurable group X is separated if for each x ≠ e X in X, there is a measurableE ⊂ X of finite, positive measure such that µ(E△xE) > 0.Theorem 6.10 (Steinhaus Theorem – cf. Comfort [Com, Th. 4.6 p. 1175]). Let X be alocally compact topological group which is separated under its Haar measure. For measurableA having positive finite Haar measure, the sets AA −1 and A −1 A have non-emptyinterior.Proof. For X separated, we recall (see [Hal-M, Sect. 62] and [We]) that the Weil topologyon X, under which X is a topological group, is generated by the neighbourhood base ate X comprising sets of the form N E,ε := {x ∈ X : µ(E△xE) < ε}, with ɛ > 0 andE measurable and of finite positive measure. Recall from [Hal-M, Sect. 62] the followingresults: (Th. F ) a measurable set with non-empty interior has positive measure; (Th. A) aset of positive measure contains a set of the form GG −1 , with G measurable and of finite,positive measure; and (Th. B) for such G, N Gε ⊆ GG −1 for all small enough ε > 0. Thusa measurable set has positive measure iff it is non-meagre in the Weil topology. Thus if Ais measurable and has positive measure it is non-meagre in the Weil topology. Moreover,by [Hal-M] Sect 61, Sect. 62 Ths. A and B, the metric open sets of X are generatedby sets of the form N E,ε for some Borelian-(K) set E of positive, finite measure. Bythe Piccard-Pettis Theorem, Th. 6.3 (from the Category Embedding Theorem, Th. 6.1)AA −1 contains a non-empty Weil neighbourhood N E,ε .Remark. See Section 7 below for an alternative proof via the density topology drawingon Mueller’s Haar-measure density theorem [Mue] and a category-measure theorem ofMartin [Mar] (and also for extensions to products AB). The following theorem has two