Abstract:

Let $A$ denote a $\sigma$-algebra of subsets of a set $\Omega, \; G$ a finite group of $(A,A)$-measurable transformations $g:\Omega\to\Omega, \; F(G)$ the set consisting of all $\omega \in \Omega$ such that $g(\omega)=\omega, \; g\in G$, is fulfilled, and let $B(G,A)$ stand for the $\sigma$-algebra consisting of all sets $A\in A$ satisfying $g(A)=A, \; g\in G$. Under the assumption $f(B)\in A^{\vert G\vert }, \; B\in B(G,A)$, for $f:\Omega\to\Omega^{\vert G\vert }$ defined by $f(\omega)= (g_1(\omega),\ldots,g_{\vert G\vert }(\omega)), \; \omega \in \Omega, \; \{g_1,\ldots,g_{\vert G\vert }\}=G$, where $\vert G\vert $ stands for the number of elements of $G, \; \Omega^{\vert G\vert }$ for the $\vert G\vert $-fold Cartesian product of $\Omega$, and $A^{\vert G\vert }$ for the $\vert G\vert $-fold direct product of $A$, it is shown that a probability measure $P$ on $A$ is uniquely determined among all probability measures on $A$ by its restriction to $B(G,A)$ if and only if $P^\ast(F(G))=1$ holds true and that $F(G)\in A $ is equivalent to the property of $A$ to separate all points $\omega _1,\omega_2\in F(G), \; \omega_1\not= \omega_2$, and $\omega\in F(G), \; \omega'\notin F(G)$, by a countable system of sets contained in $A$. The assumption $f(B)\in A^{\vert G\vert }, \; B\in B(G,A)$, is satisfied, if $\Omega$ is a Polish space and $A$ the corresponding Borel $\sigma$-algebra.