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 $heta\; in\; mathrm\{Lin\}\; (E;\; G)$, we say that $heta$ is "radonifying" if the push forward CSM (see below) $left\{\; left.\; left(\; heta\_\{*\}\; (mu\_\{cdot\})\; ight)\_\{S\}\; ight|\; S\; in\; mathcal\{A\}\; (G)\; ight\}$ on $G$ "is" a measure, i.e. there is a measure $u$ on $G$ such that::$left(\; heta\_\{*\}\; (mu\_\{cdot\})\; ight)\_\{S\}\; =\; S\_\{*\}\; (\; u)$for each $S\; in\; mathcal\{A\}\; (G)$, where $S\_\{*\}\; (\; u)$ is the usual push forward of the measure $u$ by the linear map $S\; :\; G\; o\; F\_\{S\}$.

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::$left\{\; left.\; left(\; heta\_\{*\}\; (mu\_\{cdot\})\; ight)\_\{S\}\; ight|\; S\; in\; mathcal\{A\}\; (G)\; ight\}$is defined by::$left(\; heta\_\{*\}\; (mu\_\{cdot\})\; ight)\_\{S\}\; =\; mu\_\{S\; circ\; heta\}$if the composition $S\; circ\; heta\; :\; E\; o\; F\_\{S\}$ is surjective. If $S\; circ\; heta$ is not surjective, let $ilde\{F\}$ be the image of $S\; circ\; heta$, let $i\; :\; ilde\{F\}\; o\; F\_\{S\}$ be the inclusion map, and define::$left(\; heta\_\{*\}\; (mu\_\{cdot\})\; ight)\_\{S\}\; =\; i\_\{*\}\; left(\; mu\_\{Sigma\}\; ight)$,where $Sigma\; :\; E\; o\; ilde\{F\}$ (so $Sigma\; in\; mathcal\{A\}\; (E)$) is such that $i\; circ\; Sigma\; =\; S\; circ\; heta$.

ee also

* Abstract Wiener space