Radonifying function

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

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

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

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

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

is defined by

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

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

::$\backslash left(\; \backslash theta\_\{*\}\; (\backslash mu\_\{\backslash cdot\})\; \backslash right)\_\{S\}\; =\; i\_\{*\}\; \backslash left(\; \backslash mu\_\{\backslash Sigma\}\; \backslash right)$,

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

• {{annotated link|Abstract Wiener space}}
• {{annotated link|Classical Wiener space}}
• {{annotated link|Sazonov's theorem}}

{{reflist}}

{{DEFAULTSORT:Radonifying Function}}

{{Measure theory}}

{{Analysis in topological vector spaces}}

{{Functional analysis}}

Category:Banach spaces

Category:Measure theory

Category:Types of functions