Radonifying function explained

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 l{A} (E) \} on E and a continuous linear map \theta \in Lin (E; G) , we say that \theta is radonifying if the push forward CSM (see below) \left\{ \left. \left( \theta _{*} (\mu _{⋅} ) \right) _{S} \right| S \in l{A} (G) \right\} on G "is" a measure, i.e. there is a measure \nu on G such that\left( \theta _{*} (\mu _{⋅} ) \right) _{S} = S _{*} (\nu) for each S \in l{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 l{A} (G) be surjective, the definition of the push forward for a CSM requires careful attention. The CSM\left\{ \left. \left( \theta _{*} (\mu _{⋅} ) \right) _{S} \right| S \in l{A} (G) \right\} is defined by\left( \theta _{*} (\mu _{⋅} ) \right) _{S} = \mu _{S} 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 _{⋅} ) \right) _{S} = i _{*} \left( \mu _{\Sigma} \right) ,where \Sigma : E \to \tilde{F} (so \Sigma \in l{A} (E) ) is such that i \circ \Sigma = S \circ \theta