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.
Given two separable Banach spaces
and a continuous linear map
θ ∈
θ
is radonifying if the push forward CSM (see below)
θ
ν
( ν )
is the usual push forward of the measure
by the linear map
Because the definition of a CSM on
requires that the maps in
be surjective, the definition of the push forward for a CSM requires careful attention.
The CSM is defined by if the composition
∘ θ :
∘ θ
∘ θ
be the inclusion map, and define where
∘ θ