The Steinhaus-Weil property: its converse, subcontinuity and Solecki amenability

Abstract

The Steinhaus-Weil theorem that concerns us here is the simple, or classical, 'interior-points' property -that in a Polish topological group a nonnegligible set B has the identity as an interior point of B −1 B. There are various converses; the one that mainly concerns us is due to Simmons and Mospan. Here the group is locally compact, so we have a Haar reference measure η. The Simmons-Mospan theorem states that a (regular Borel) measure has such a Steinhaus-Weil property if and only if it is absolutely continuous with respect to the Haar measure. In Part I (Propositions 1-7, Theorems 1-4) we exploit the connection between the interior-points property and a selective form of infinitesimal invariance afforded by a certain family of selective reference measures σ, drawing on Solecki's amenability at 1 (and using Fuller's notion of subcontinuity). In Part II (Propositions 8, 9, Theorems 5, 6) we develop a number of relatives of the Simmons-Mospan theorem. In Part III (Theorems 7, 8) we link this with topologies of Weil type. We close in Part IV with Propositions 12-13 and Theorem B -concerning the 'composite interior-point' property (of AB −1 ) and the Borell 'relative interior-point' property (relative to the Cameron-Martin space) -and complements.

The Gołąb-Schinzel and Goldie functional equations in Banach algebras

“…The two particular cases asserted follow from the Interior-point Theorem for category (Steinhaus-Piccard-Pettis theorem [Oxt,Th. 4.8] or [BinO2]): indeed, G S , as an open subset of a Banach space, is analytic, so S(G S ), being the continuous image of an analytic set, is analytic, so has the Baire property, by Nikodym's theorem [Rog]. So 1 A ∈ int(S(G S )S(G S ) −1 )⊆S(G S ), the latter inclusion holding because S(G S ) is a group.…”