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.

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

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

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

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

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

is defined by

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

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

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

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

• Abstract Wiener space – Mathematical construction relating to infinite-dimensional spaces
• Classical Wiener space – Space of stochastic processes
• Sazonov's theorem

• Banach spaces
• Measure theory
• Types of functions

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