Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus–Weil theorem

Abstract

The theme here is category-measure duality, in the context of a topological group. One can often handle the (Baire) category case and the (Lebesgue, or Haar) measure cases together, by working bi-topologically: switching between the original topology and a suitable refinement (a density topology). This prompts a systematic study of such density topologies, and the corresponding σ-ideals of negligibles. Such ideas go back to Weil's classic book, and to Hashimoto's ideal topologies. We make use of group norms, which cast light on the interplay between the group and measure structures. The Steinhaus-Weil interior-points theorem ('on AA −1 ') plays a crucial role here; so too does its converse, the Simmons-Mospan theorem.

Beyond Haar and Cameron-Martin: The Steinhaus support

“…In Theorem 3.4 below we strengthen this conclusion to yield the measure-theoretic 'Kemperman property', introduced in [Kem], as in [BinO4]. (One would expect this to imply shift-compactness for 'G-shifts of H', as indeed is so -see Th.…”

“…In §5 we exhibit a link between the group context and the classical Cameron-Martin theory above by verifying that a divisible abelian group with an N-homogeneous group-norm (below) is in fact a topological vector space. In §6 we extend our usage in §3 of the notions of subcontinuity and selective subcontinuity of a measure (introduced in [BinO4,7]), and in Theorem 6.1 establish the key property of a Solecki reference measure. Then in §7 for a given Polish group G and 'sufficiently subcontinuous' measure µ we construct a corresponding subset H(µ), which together with G and µ forms a Steinhaus triple (possibly 'selective': see below).…”

“…Recall from the companion paper [BinO7] (cf. [BinO4]) the following definition, already used in Th. 3.4 above.…”

General regular variation, Popa groups and quantifier weakening

“…8] -such sets can be 'thinned out', i.e. extracted as subsets of a second-category set, using separability or by reference to the Banach Category Theorem [Oxt,Ch.16]); (iii) Σ the Cantor 'excluded middle-thirds' subset of [0, 1] (since Σ + Σ = [0, 2]); (iv) Σ universally measurable and open in the ideal topology ( [LukMZ], [BinO9]) generated by omitting Haar null sets (by the Christensen-Solecki Interior-points Theorem of [Sol]); (v) Σ a Borel subset of a Polish abelian group and and open in the ideal topology generated by omitting Haar meagre sets in the sense of Darji [Dar] (by Jabłońska's generalization of the Piccard Theorem, [Jab1, Th. 2], cf.…”

Variants on the Berz sublinearity theorem

“…8] -such sets can be 'thinned out', i.e. extracted as subsets of a second-category set, using separability or by reference to the Banach Category Theorem [Oxt,Ch.16]); (iii) Σ the Cantor 'middle-thirds excluded' subset of [0, 1] (since Σ + Σ = [0, 2]); (iv) Σ universally measurable and open in the ideal topology ( [LukMZ], [BinO4]) generated by omitting Haar null sets (by the Christensen-Solecki Interior-points Theorem of [Chr1,2] and [Sol]); (v) Σ a Borel subset of a Polish abelian group and and open in the ideal topology generated by omitting Haar meagre sets in the sense of Darji [Dar] (by Jab lońska's generalization of the Piccard Theorem, [Jab1, Th.2], cf. [Jab3], and since the Haar-meagre sets form a σ-ideal [Dar,Th.…”