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 $\{\mu _{T}|T\in {\mathcal {A}}(E)\}$ 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

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 $i\circ \Sigma =S\circ \theta$ .

References

v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
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
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

Definition

Push forward of a CSM

See also

References