Radonify

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.

Given two separable Banach spaces ${\displaystyle E}$ and ${\displaystyle G}$, a CSM ${\displaystyle \{\mu _{T}|T\in {\mathcal {A}}(E)\}}$ on ${\displaystyle E}$ and a continuous linear map ${\displaystyle \theta \in \mathrm {Lin} (E;G)}$, we say that ${\displaystyle \theta }$ is radonifying if the push forward CSM (see below) ${\displaystyle \left\{\left.\left(\theta _{*}(\mu _{\cdot })\right)_{S}\right|S\in {\mathcal {A}}(G)\right\}}$ on ${\displaystyle G}$ "is" a measure, i.e. there is a measure ${\displaystyle \nu }$ on ${\displaystyle G}$ such that

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