Radonifying function: Difference between revisions

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{{Unreferenced|date=December 2009}}

In [[measure theory]], a '''radonifying function''' (ultimately named after [[Johann Radon]]) between [[measurable space]]s 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 space|separable]] [[Banach space]]s <math>E</math> and <math>G</math>, a CSM <math>\{ \mu_{T} | T \in \mathcal{A} (E) \}</math> on <math>E</math> and a [[continuous function|continuous]] [[linear map]] <math>\theta \in \mathrm{Lin} (E; G)</math>, we say that <math>\theta</math> is ''radonifying'' if the push forward CSM (see below) <math>\left\{ \left. \left( \theta_{*} (\mu_{\cdot}) \right)_{S} \right| S \in \mathcal{A} (G) \right\}</math> on <math>G</math> "is" a measure, i.e. there is a measure <math>\nu</math> on <math>G</math> such that

Given two [[separable]] [[Banach space]]s <math>E</math> and <math>G</math>, a CSM <math>\{ \mu_{T} | T \in \mathcal{A} (E) \}</math> on <math>E</math> and a [[continuous]] [[linear map]] <math>\theta \in \mathrm{Lin} (E; G)</math>, we say that <math>\theta</math> is ''radonifying'' if the push forward CSM (see below) <math>\left\{ \left. \left( \theta_{*} (\mu_{\cdot}) \right)_{S} \right| S \in \mathcal{A} (G) \right\}</math> on <math>G</math> "is" a measure, i.e. there is a measure <math>\nu</math> on <math>G</math> such that

::<math>\left( \theta_{*} (\mu_{\cdot}) \right)_{S} = S_{*} (\nu)</math>

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

==Push forward of a CSM==

Because the definition of a CSM on <math>G</math> requires that the maps in <math>\mathcal{A} (G)</math> be [[surjective]], the definition of the push forward for a CSM requires careful attention. The CSM

::<math>\left\{ \left. \left( \theta_{*} (\mu_{\cdot}) \right)_{S} \right| S \in \mathcal{A} (G) \right\}</math>

is defined by

::<math>\left( \theta_{*} (\mu_{\cdot}) \right)_{S} = \mu_{S \circ \theta}</math>

if the [[Function_compositionFunction composition|composition]] <math>S \circ \theta : E \to F_{S}</math> is surjective. If <math>S \circ \theta</math> is not surjective, let <math>\tilde{F}</math> be the image of <math>S \circ \theta</math>, let <math>i : \tilde{F} \to F_{S}</math> be the [[inclusion map]], and define

::<math>\left( \theta_{*} (\mu_{\cdot}) \right)_{S} = i_{*} \left( \mu_{\Sigma} \right)</math>,

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

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==See also==

* [[{{annotated link|Abstract Wiener space]]}}

* {{annotated link|Classical Wiener space}}

* {{annotated link|Sazonov's theorem}}

==References==

{{reflist}}

{{DEFAULTSORT:Radonifying Function}}

[[Category:{{Measure theory]]}}

{{Analysis in topological vector spaces}}

{{Functional analysis}}

[[Category:Measure theory]]

[[Category:Banach spaces]]

[[Category:Measure theory]]

[[Category:Types of functions]]