Multihomogeneous Measures and Stochastic Polar Representations

Let $q\in \mathbb{N}$, let $G=(0,\mathrm{\infty }{)}^q$, and let $S:G\times E⟶E,(r,x)⟼S_{r}x$ be a jointly measurable left action on an arbitrary measurable space $(E,\mathcal{E})$. For $\alpha =({\alpha }_1,\dots ,{\alpha }_q)\in (0,\mathrm{\infty }{)}^q$ set ${\chi }_{\alpha }(r)={\prod }_{i=1}^q{r}_i^{{\alpha }_i}\mathrm{.}$ We study nonzero $\sigma$-finite measures $\nu$ satisfying $\nu (S_{r}A)={\chi }_{\alpha }(r{)}^{-1}\nu (A),r\in G,A\in \mathcal{E}\mathrm{.}$ Motivated by the scalar case $q=1$ studied in [1] we derive equivalent conditions for the existence of an $E$-valued random element $Z$ such that $\nu (A)=\mathbb{E}\{{\int }_G{\mathbb{I}}_A(S_{r}Z){\prod }_{i=1}^q{\alpha }_i{r}_i^{-{\alpha }_i-1}d{r}_i\},A\in \mathcal{E}\mathrm{.}$ We also characterise when two random elements generate the same homogeneous measure, using multihomogeneous moments and, after fixing an admissible product gauge, weighted transverse measures. When the corresponding weighted transverse measure is finite, tilting and gauge normalisation produce a canonical representer, unique in law on the prescribed gauge shell. Finally, we characterise stationarity under an action commuting with $S$ and construct positive semidefinite tail-overlap kernels directly from $\nu$.