Radonifying function

Definition

Given two separable Banach spaces E and G , a CSM { μ T | T ∈ A ( E ) } on E and a continuous linear map θ ∈ L i n ( E ; G ) , we say that θ is radonifying if the push forward CSM (see below) { ( θ ∗ ( μ ⋅ ) ) S | S ∈ A ( G ) } on G "is" a measure, i.e. there is a measure ν on G such that

for each S ∈ A ( G ) , where S ∗ ( ν ) is the usual push forward of the measure ν by the linear map S : G → 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

is defined by

if the composition S ∘ θ : E → F S is surjective. If S ∘ θ is not surjective, let F ~ be the image of S ∘ θ , let i : F ~ → F S be the inclusion map, and define

where Σ : E → F ~ (so Σ ∈ A ( E ) ) is such that i ∘ Σ = S ∘ θ .