Radonifying function
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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.
[edit] Definition
Given two separable Banach spaces E and G, a CSM 
on E and a continuous linear map 
, we say that θ is radonifying if the push forward CSM (see below) 
on G "is" a measure, i.e. there is a measure ν on G such that
for each 
, where S * (ν) is the usual push forward of the measure ν by the linear map 
.
[edit] Push forward of a CSM
Because the definition of a CSM on G 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 
is surjective. If 
is not surjective, let 
be the image of , let 
be the inclusion map, and define
-
,
where 
(so 
) is such that 
.
