Beyond Haar and Cameron-Martin: the Steinhaus Support
Beyond Haar and Cameron-Martin: the Steinhaus support N. H. Bingham and A. J. Ostaszewski In memoriam Herbert Heyer 1936-2018 Abstract. Motivated by a Steinhaus-like interior-point property involving the Cameron-Martin space of Gaussian measure theory, we study a group- theoretic analogue, the Steinhaus triple (H,G,µ), and construct a Stein- haus support, a Cameron-Martin-like subset, H(µ) in any Polish group G corresponding to ‘sufficiently subcontinuous’ measures µ, in particular for ‘Solecki-type’ reference measures. Key-words. Cameron-Martin space, Gaussian measures, relativized Stein- haus (interior-point) property, Steinhaus triple, Steinhaus support, amenabil- ity at 1, relative quasi-invariance, abstract Wiener space, measure subconti- nuity. AMS Classification Primary 22A10, 43A05; 60-B-15; Secondary 28C10, 60-B-05, 60-B-11. 1 Introduction For many purposes, one needs a reference measure. In discrete situations such as the integers Z, one has counting measure. In Euclidean space Rd, one has Lebesgue measure. In locally compact groups, one has Haar measure. In infinite-dimensional settings such as Hilbert space, one has neither local arXiv:1805.02325v3 [math.FA] 14 Mar 2019 compactness nor Haar measure. Here various possibilities arise, some patho- logical – which we avoid by restricting to Radon measures (below). One is to use Christensen’s concept of Haar-null sets (below), even though there is no Haar measure; see [Chr1,2], Solecki [Sol], and the companion paper to this, [BinO7]. Another is to use Gaussian measures; for background see e.g. Bogachev [Bog1,3], Kuo [Kuo] and for Gaussian processes, Lifshits [Lif], Marcus and Rosen [MarR], Ibragimov and Rozanov [IbrR].
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