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.
Definition 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}} .
Push forward of a CSM 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 }}
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