Radonifying function

Push forward of a CSM

Because the definition of a CSM on ${\displaystyle G}$ requires that the maps in ${\displaystyle \mathcal{A}(G)}$ be surjective, the definition of the push forward for a CSM requires careful attention. The CSM

${\displaystyle \left\{{\left({\theta }_{\ast }({\mu }_{\cdot })\right)}_S|S\in \mathcal{A}(G)\right\}}$

is defined by

${\displaystyle {\left({\theta }_{\ast }({\mu }_{\cdot })\right)}_S={\mu }_{S\circ \theta }}$

if the composition ${\displaystyle S\circ \theta :E\to F_S}$ is surjective. If ${\displaystyle S\circ \theta }$ is not surjective, let ${\displaystyle \tilde{F}}$ be the image of ${\displaystyle S\circ \theta }$, let ${\displaystyle i:\tilde{F}\to F_S}$ be the inclusion map, and define

${\displaystyle {\left({\theta }_{\ast }({\mu }_{\cdot })\right)}_S=i_{\ast }\left({\mu }_{\mathrm{\Sigma }}\right)}$,

where ${\displaystyle \mathrm{\Sigma }:E\to \tilde{F}}$ (so ${\displaystyle \mathrm{\Sigma }\in \mathcal{A}(E)}$) is such that ${\displaystyle i\circ \mathrm{\Sigma }=S\circ \theta }$.