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 [math]\displaystyle{ E }[/math] and [math]\displaystyle{ G }[/math], a CSM [math]\displaystyle{ \{ \mu_{T} | T \in \mathcal{A} (E) \} }[/math] on [math]\displaystyle{ E }[/math] and a continuous linear map [math]\displaystyle{ \theta \in \mathrm{Lin} (E; G) }[/math], we say that [math]\displaystyle{ \theta }[/math] is radonifying if the push forward CSM (see below) [math]\displaystyle{ \left\{ \left. \left( \theta_{*} (\mu_{\cdot}) \right)_{S} \right| S \in \mathcal{A} (G) \right\} }[/math] on [math]\displaystyle{ G }[/math] "is" a measure, i.e. there is a measure [math]\displaystyle{ \nu }[/math] on [math]\displaystyle{ G }[/math] such that
- [math]\displaystyle{ \left( \theta_{*} (\mu_{\cdot}) \right)_{S} = S_{*} (\nu) }[/math]
for each [math]\displaystyle{ S \in \mathcal{A} (G) }[/math], where [math]\displaystyle{ S_{*} (\nu) }[/math] is the usual push forward of the measure [math]\displaystyle{ \nu }[/math] by the linear map [math]\displaystyle{ S : G \to F_{S} }[/math].
Push forward of a CSM
Because the definition of a CSM on [math]\displaystyle{ G }[/math] requires that the maps in [math]\displaystyle{ \mathcal{A} (G) }[/math] be surjective, the definition of the push forward for a CSM requires careful attention. The CSM
- [math]\displaystyle{ \left\{ \left. \left( \theta_{*} (\mu_{\cdot}) \right)_{S} \right| S \in \mathcal{A} (G) \right\} }[/math]
is defined by
- [math]\displaystyle{ \left( \theta_{*} (\mu_{\cdot}) \right)_{S} = \mu_{S \circ \theta} }[/math]
if the composition [math]\displaystyle{ S \circ \theta : E \to F_{S} }[/math] is surjective. If [math]\displaystyle{ S \circ \theta }[/math] is not surjective, let [math]\displaystyle{ \tilde{F} }[/math] be the image of [math]\displaystyle{ S \circ \theta }[/math], let [math]\displaystyle{ i : \tilde{F} \to F_{S} }[/math] be the inclusion map, and define
- [math]\displaystyle{ \left( \theta_{*} (\mu_{\cdot}) \right)_{S} = i_{*} \left( \mu_{\Sigma} \right) }[/math],
where [math]\displaystyle{ \Sigma : E \to \tilde{F} }[/math] (so [math]\displaystyle{ \Sigma \in \mathcal{A} (E) }[/math]) is such that [math]\displaystyle{ i \circ \Sigma = S \circ \theta }[/math].
See also
References
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