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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.
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
where (so ) is such that .