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 $\$ on $E$ and a continuous linear map $\theta \in \mathrm (E; G)$, we say that $\theta$ is ''radonifying'' if the push forward CSM (see below) $\left\$ on $G$ "is" a measure, i.e. there is a measure $\nu$ on $G$ such that
::$\left( \theta_ (\mu_) \right)_ = S_ (\nu)$
for each $S \in \mathcal (G)$, where $S_ (\nu)$ is the usual push forward of the measure $\nu$ by the linear map $S : G \to F_$.

Push forward of a CSM

Because the definition of a CSM on $G$ requires that the maps in $\mathcal (G)$ be surjective, the definition of the push forward for a CSM requires careful attention. The CSM
::$\left\$
is defined by
::$\left( \theta_ (\mu_) \right)_ = \mu_$
if the composition $S \circ \theta : E \to F_$ is surjective. If $S \circ \theta$ is not surjective, let $\tilde$ be the image of $S \circ \theta$, let $i : \tilde \to F_$ be the inclusion map, and define
::$\left( \theta_ (\mu_) \right)_ = i_ \left( \mu_ \right)$,
where $\Sigma : E \to \tilde$ (so $\Sigma \in \mathcal (E)$) is such that $i \circ \Sigma = S \circ \theta$.

See also

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References

{{Functional analysis

Banach spaces
Measure theory
Types of functions