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

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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

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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

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References

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