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.

Definition

Given two separable Banach spaces E and G, a CSM on E and a continuous linear map, we say that \theta is radonifying if the push forward CSM (see below) on G "is" a measure, i.e. there is a measure \nu on G such that for each, where S_{*} (\nu) is the usual push forward of the measure \nu by the linear map.

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 \tilde{F} be the image of, let be the inclusion map, and define where (so ) is such that.