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 $$\theta \in \mathrm{Lin} (E; G)$$, we say that $$\theta$$ is radonifying if the push forward CSM (see below) $$\left\{ \left. \left( \theta_{*} (\mu_{\cdot}) \right)_{S} \right| S \in \mathcal{A} (G) \right\}$$ on $$G$$ "is" a measure, i.e. there is a measure $$\nu$$ on $$G$$ such that
 * $$\left( \theta_{*} (\mu_{\cdot}) \right)_{S} = S_{*} (\nu)$$

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

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
 * $$\left\{ \left. \left( \theta_{*} (\mu_{\cdot}) \right)_{S} \right| S \in \mathcal{A} (G) \right\}$$

is defined by
 * $$\left( \theta_{*} (\mu_{\cdot}) \right)_{S} = \mu_{S \circ \theta}$$

if the composition $$S \circ \theta : E \to F_{S}$$ is surjective. If $$S \circ \theta$$ is not surjective, let $$\tilde{F}$$ be the image of $$S \circ \theta$$, let $$i : \tilde{F} \to F_{S}$$ be the inclusion map, and define
 * $$\left( \theta_{*} (\mu_{\cdot}) \right)_{S} = i_{*} \left( \mu_{\Sigma} \right)$$,

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