Measurable finite-group permutation setups

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment (Statistical experiments), let $\mathrm{\varnothing }\mathrm{\ne }{\mathrm{\Theta }}_0\subseteq \mathrm{\Theta }$, and let $G$ be a finite group (Group and abelian group) acting on $\mathcal{X}$ by a left action (Left group actions, transitive actions, and faithful actions). Require that each map $x\mapsto g\cdot x$ is measurable from $(\mathcal{X},\mathcal{A})$ to itself and that $T:\mathcal{X}\to \mathbb{R}$ is measurable (A measurable function between measurable spaces).

For $\theta \in {\mathrm{\Theta }}_0$, the null law is invariant under the action when $P_{\theta }(\{x:g\cdot x\in A\})=P_{\theta }(A)$ for every $g\in G$ and $A\in \mathcal{A}$. Equivalently, $g\cdot X$ and $X$ have the same law when $X\sim P_{\theta }$. The setup defines the action and score whether or not this invariance condition holds; the validity theorem below assumes it for every null parameter.

The data space need not be finite, and the action need not be free. Group elements, including distinct elements that send $x$ to the same point, remain distinct transformations in every group average. On a finite data space with the full power-set sigma-algebra, if every null law is invariant, the setup specializes to Finite-group randomization setups.