Small-ball MAP conventions and Cameron–Martin relative modes

Let $B$ be a separable Banach space with norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_B$, and let $\mathrm{\Pi }$ be a Borel probability measure on $B$. For $z\in B$ and $r>0$, write ${\bar{B}}_B(z,r)=\{u\in B:\mathrm{\parallel }u-z{\mathrm{\parallel }}_B\le r\}$ and $m_r(z)=\mathrm{\Pi }({\bar{B}}_B(z,r))$. A strong small-ball MAP is a point $z\in B$ for which $\frac{m_r(z)}{{\sup }_{x\in B}{m}_r(x)}⟶1(r\downarrow 0)\mathrm{.}$ Assume AC for the covariance-induced Cameron–Martin notation. In this Gaussian branch, regard $B$ as a real Banach space, restricting scalars if $B$ is complex; $B^{\ast }$ denotes its continuous real dual, and Gaussianity is tested against these real functionals. The norm, Borel sets, and balls are unchanged by this restriction. If $\mathrm{\Pi }$ is a posterior law formed by weighting a centered Borel Gaussian prior ${\mathrm{\Pi }}_0$ on $B$, let $C\mathrm{\ell }={\int }_B\mathrm{\ell }(u)u{\mathrm{\Pi }}_0(du)$ be its covariance operator and let $E$ be the completion of $B^{\ast }\mathrm{/}\ker C$ with inner product $⟨[\mathrm{\ell }],[m]{⟩}_E=m(C\mathrm{\ell })$; the canonical injection $i:E\to B$ extends $[\mathrm{\ell }]\mapsto C\mathrm{\ell }$. A Cameron–Martin ratio mode is a point $ih\in iE$ such that, for every $k\in E$, the denominator is positive for all sufficiently small $r$ and ${\lim }_{r\downarrow 0}\frac{m_r(ih)}{m_r(ik)}$ exists and is at least $1$. This pairwise comparison over fixed Cameron–Martin centers does not assert the global strong small-ball MAP condition.

For every $r>0$, ${\sup }_{x\in B}{m}_r(x)>0$: a countable dense subset gives a countable cover of $B$ by radius-$r$ open balls, each contained in its corresponding closed ball, and countable subadditivity rules out all such closed balls having zero probability.

the posterior is the normalized likelihood-potential weighting of a prior

a Borel prior is concentrated on its declared forward-map domain

the covariance-dual quotient, Cameron–Martin norm, and canonical injection

a separable space has an at most countable dense subset

countable subadditivity prevents a probability space from being a countable union of null balls

the Axiom of Choice assumed for the Gaussian covariance/RKHS branch

a Borel probability is a probability measure on the Borel sigma-algebra

a Banach space is complete in its norm metric

restriction to real scalars preserves the norm and metric of a complex Banach space