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.

Definition[edit]

Given two separable Banach spaces and , a CSM on and a continuous linear map , we say that is radonifying if the push forward CSM (see below) 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[edit]

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 (so ) is such that .

See also[edit]