Multihomogeneous Measures and Stochastic Polar Representations

Let \(q\in\mathbb N\), let \(G=(0,\infty)^q\), and let $ S:G\times E\longrightarrow E, (r,x)\longmapsto S_rx $ be a jointly measurable left action on an arbitrary measurable space \((E,\mathcal E)\). For \(α=(α_1,\ldots,α_q)\in(0,\infty)^q\) set $ χ_α(r)=\prod_{i=1}^q r_i^{α_i}. $ We study nonzero \