Radonifying function

In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the second space. It acquired its name because the pushforward measure on the second space was historically thought of as a Radon measure.

Definition

Given two separable Banach spaces $E$ and $G$, a CSM $\{{\mu }_T|T\in {𝒜}(E)\}$ on $E$ and a continuous linear map $\theta \in \mathrm{L}\mathrm{i}\mathrm{n}(E;G)$, we say that $\theta$ is radonifying if the push forward CSM (see below) $\{{({\theta }_{*}({\mu }_{\cdot }))}_S|S\in {𝒜}(G)\}$ on $G$ "is" a measure, i.e. there is a measure $\nu$ on $G$ such that

${({\theta }_{*}({\mu }_{\cdot }))}_S=S_{*}(\nu )$

for each $S\in {𝒜}(G)$, where $S_{*}(\nu )$ is the usual push forward of the measure $\nu$ by the linear map $S:G\to F_S$.

Push forward of a CSM

Because the definition of a CSM on $G$ requires that the maps in ${𝒜}(G)$ be surjective, the definition of the push forward for a CSM requires careful attention. The CSM

$\{{({\theta }_{*}({\mu }_{\cdot }))}_S|S\in {𝒜}(G)\}$

is defined by

${({\theta }_{*}({\mu }_{\cdot }))}_S={\mu }_{S\circ \theta }$

if the composition $S\circ \theta :E\to F_S$ is surjective. If $S\circ \theta$ is not surjective, let $\tilde{F}$ be the image of $S\circ \theta$, let $i:\tilde{F}\to F_S$ be the inclusion map, and define

${({\theta }_{*}({\mu }_{\cdot }))}_S=i_{*}\left({\mu }_{\mathrm{\Sigma }}\right)$,

where $\mathrm{\Sigma }:E\to \tilde{F}$ (so $\mathrm{\Sigma }\in {𝒜}(E)$) is such that $i\circ \mathrm{\Sigma }=S\circ \theta$.

See also

• Abstract Wiener space
• Classical Wiener space
• Sazonov's theorem