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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