Radonifying function

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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[edit]

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[edit]

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.

See also[edit]