Radonifying function

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 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

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

This article is issued from Wikipedia - version of the 11/12/2011. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.