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 ${μ_{T} | T \in 𝒜 (E)}$ on $E$ and a continuous linear map $θ \in L i n (E; G)$ , we say that $θ$ is radonifying if the push forward CSM (see below) ${{(θ_{*} (μ_{\cdot}))}_{S} | S \in 𝒜 (G)}$ on $G$ "is" a measure, i.e. there is a measure $ν$ on $G$ such that

{(θ_{*} (μ_{\cdot}))}_{S} = S_{*} (ν)

for each $S \in 𝒜 (G)$ , where $S_{*} (ν)$ is the usual push forward of the measure $ν$ 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 $𝒜 (G)$ be surjective, the definition of the push forward for a CSM requires careful attention. The CSM

{{(θ_{*} (μ_{\cdot}))}_{S} | S \in 𝒜 (G)}

is defined by

{(θ_{*} (μ_{\cdot}))}_{S} = μ_{S \circ θ}

if the composition $S \circ θ : E \to F_{S}$ is surjective. If $S \circ θ$ is not surjective, let $\tilde{F}$ be the image of $S \circ θ$ , let $i : \tilde{F} \to F_{S}$ be the inclusion map, and define

{(θ_{*} (μ_{\cdot}))}_{S} = i_{*} (μ_{Σ})

,

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

References

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v t e Analysis in topological vector spaces
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Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
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Radonifying function

Contents

Definition

Push forward of a CSM

See also

References

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