MONOGENICITY OF PROBABILITY-MEASURES BASED ON MEASURABLE SETS INVARIANT UNDER FINITE-GROUPS OF TRANSFORMATIONS

Let A denote a sigma-algebra of subsets of a set Omega, G a finite gro up of (st, A)-measurable transformations g : Omega --> Omega, F(G) the set consisting of all omega is an element of Omega such that g(omega) = omega, g is an element of G, is fulfilled, and let B(G, A) stand fo r the sigma-algebra consisting of all sets A is an element of A satisf ying g(A) = A, g is an element of G. Under the assumption f(B) is an e lement of A(\G\), B is an element of B(G, A), for f : Omega --> Omega( \G\) defined by f(omega) = (g(1)(omega),...,g(\G\)(omega)), omega is a n element of Omega, {g(1),...,g(\G\)} = G, where \G\ stands for the nu mber of elements of G, Omega(\G\) for the \G\-fold Cartesian product o f Omega, and A(\G\) for the \G\-fold direct product of A, it is shown that a probability measure P on A is uniquely determined among all pro bability measures on A by its restriction to a(G, A) if and only if P (F(G)) = 1 holds true and that F(G) is an element of A is equivalent t o the property of A to separate all points omega(1), omega(2) is an el ement of F(G), omega(1) not equal omega(2), and omega is an element of F(G), omega' is not an element of F(G), by a countable system of sets contained in A. The assumption f(B) is an element of A(\G\), B is an element of B(G, A), is satisfied, if Omega is a Polish space and A the corresponding Borel sigma-algebra.