Radonifying function

In measure theory, a radonifying function between measurable spaces is one that takes a cylinder set measure 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 and, a CSM on and a continuous linear map, we say that is radonifying if the push forward CSM on "is" a measure, i.e. there is a measure on such that
for each, where is the usual push forward of the measure by the linear map.

Push forward of a CSM

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 is surjective. If is not surjective, let be the image of, let be the inclusion map, and define
where is such that.