Radonifying function

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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.

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

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

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

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

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

is defined by

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

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

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

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

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