Radonifying operator

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In measure theory, a radonifying operator (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.

Given two separable Banach spaces E {\displaystyle E} and G {\displaystyle G} , a CSM { μ T | T ∈ A ( E ) } {\displaystyle \{\mu _{T}|T\in {\mathcal {A}}(E)\}} on E {\displaystyle E} and a continuous linear map θ ∈ L i n ( E ; G ) {\displaystyle \theta \in \mathrm {Lin} (E;G)} , we say that θ {\displaystyle \theta } is radonifying if the push forward CSM (see below) { ( θ ∗ ( μ ⋅ ) ) S | S ∈ A ( G ) } {\displaystyle \left\{\left.\left(\theta _{*}(\mu _{\cdot })\right)_{S}\right|S\in {\mathcal {A}}(G)\right\}} on G {\displaystyle G} "is" a measure, i.e. there is a measure ν {\displaystyle \nu } on G {\displaystyle G} such that

( θ ∗ ( μ ⋅ ) ) S = S ∗ ( ν ) {\displaystyle \left(\theta _{*}(\mu _{\cdot })\right)_{S}=S_{*}(\nu )}

for each S ∈ A ( G ) {\displaystyle S\in {\mathcal {A}}(G)} , where S ∗ ( ν ) {\displaystyle S_{*}(\nu )} is the usual push forward of the measure ν {\displaystyle \nu } by the linear map S : G → F S {\displaystyle S:G\to F_{S}} .^[1]

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

{ ( θ ∗ ( μ ⋅ ) ) S | S ∈ A ( G ) } {\displaystyle \left\{\left.\left(\theta _{*}(\mu _{\cdot })\right)_{S}\right|S\in {\mathcal {A}}(G)\right\}}

is defined by

( θ ∗ ( μ ⋅ ) ) S = μ S ∘ θ {\displaystyle \left(\theta _{*}(\mu _{\cdot })\right)_{S}=\mu _{S\circ \theta }}

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

( θ ∗ ( μ ⋅ ) ) S = i ∗ ( μ Σ ) {\displaystyle \left(\theta _{*}(\mu _{\cdot })\right)_{S}=i_{*}\left(\mu _{\Sigma }\right)} ,

where Σ : E → F ~ {\displaystyle \Sigma :E\to {\tilde {F}}} (so Σ ∈ A ( E ) {\displaystyle \Sigma \in {\mathcal {A}}(E)} ) is such that i ∘ Σ = S ∘ θ {\displaystyle i\circ \Sigma =S\circ \theta } .

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