Beyond Haar and Cameron-Martin: the Steinhaus support

TL;DR

This paper introduces the Steinhaus support in Polish groups, extending Gaussian measure concepts to a broader class of measures and group structures, revealing new geometric and measure-theoretic properties.

Contribution

It constructs a Steinhaus support analogous to Cameron-Martin space for a wide class of measures in Polish groups, generalizing classical Gaussian measure results.

Findings

Defined the Steinhaus triple $(H,G,mbda)$ and its support $H(mbda)$

Extended the Steinhaus property to non-Gaussian, 'sufficiently subcontinuous' measures

Provided a framework for analyzing measure support in general Polish groups

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 $(H,G,\mu)$, and construct a Steinhaus support, a Cameron-Martin-like subset, $H(\mu)$ in any Polish group $G$ corresponding to `sufficiently subcontinuous' measures $\mu$, in particular for `Solecki-type' reference measures.