A Borel prior concentrated on a forward-map domain

Let $X$ be a specified real separable Banach or Hilbert parameter space with metric $d_X(x,x^{\mathrm{\prime }})=\mathrm{\parallel }x-x^{\mathrm{\prime }}{\mathrm{\parallel }}_X$ and its norm topology. Let $(Y,d_Y)$ be a metric data space equipped with a sigma-algebra ${\mathrm{\Sigma }}_Y$, let $U\subseteq X$ be a nonempty admissible truth set, and let $F:U\to Y$ be a forward map. Choose a Borel set $D\in \mathcal{B}(X)$ with $D\subseteq U$ such that the restriction $G:=F{\mathrm{\mid }}_D:D\to Y$ is measurable for the trace Borel sigma-algebra $\mathcal{B}(D):=\{D\cap A:A\in \mathcal{B}(X)\}$. A prior on this forward domain is a Borel probability measure $\mu$ on $X$ such that $\mu (D)=1$. Its restriction $\mu {\mathrm{\mid }}_D$, defined by $\mu {\mathrm{\mid }}_D(A)=\mu (A)$ for $A\in \mathcal{B}(D)$, is then a probability measure on $D$, and the induced data law is $G_{\mathrm{\#}}(\mu {\mathrm{\mid }}_D)$, defined by $G_{\mathrm{\#}}(\mu {\mathrm{\mid }}_D)(E)=(\mu {\mathrm{\mid }}_D)(G^{-1}(E))$ for $E\in {\mathrm{\Sigma }}_Y$. A Borel probability on $X$ with $\mu (D)<1$ is not a prior on this declared forward domain.

metric parameter and data spaces, a nonempty admissible truth set, and a forward map defined on that set

separable means having an at most countable dense subset

a Banach space is complete for its norm metric

a Hilbert space is a complete inner-product space

a Polish space is separable and completely metrizable

the Borel sigma-algebra is generated by open sets

a measurable space is a set equipped with a sigma-algebra

a probability measure has total mass one

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

measurability is defined by measurable inverse images

Dashti and Stuart, §1.3 and §3.2, full text

Giné and Nickl, §§1.1.2 and 1.2.2, full book