Beyond Haar and Cameron-Martin: the Steinhaus support

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.