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 $\{\mu _{T}|T\in {\mathcal {A}}(E)\}$ on $E$ and a continuous linear map $\theta \in \mathrm {Lin} (E;G)$, we say that $\theta$ is radonifying if the push forward CSM (see below) $\left\{\left.\left(\theta _{*}(\mu _{\cdot })\right)_{S}\right|S\in {\mathcal {A}}(G)\right\}$ on $G$ "is" a measure, i.e. there is a measure $\nu$ on $G$ such that

$\left(\theta _{*}(\mu _{\cdot })\right)_{S}=S_{*}(\nu )$

for each $S\in {\mathcal {A}}(G)$, 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 ${\mathcal {A}}(G)$ be surjective, the definition of the push forward for a CSM requires careful attention. The CSM

$\left\{\left.\left(\theta _{*}(\mu _{\cdot })\right)_{S}\right|S\in {\mathcal {A}}(G)\right\}$

is defined by

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

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

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

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

See also

• Abstract Wiener space