Continuity of quasi-invariant measures and zero-one laws on groups
Let \((X,{\mathcal B},\mu)\) be a probability space, and let \(G\) be a measurable action group on \(X\). The first aim of this paper is to give several equivalent conditions for \(\mu \ll m_ G*\mu\), where \(m_ G\) stands for the left Haar measure on \(G\). Another interesting result is as follows: Let \({\mathbf X} = (X_ n)_{n \geq 1}\) be a real random sequence, let \(\mu\) be the distribution of \({\mathbf X}\), and let \(\mu_ n\) stand for the distribution of \((X_ 1, \dots, X_ n)\). If the tail \(\sigma\)-algebra of \({\mathbf X}\) is trivial, and \(\mu_ n \ll \lambda_ n\), where \(\lambda_ n\) denotes the Lebesgue measure on \(R^ n\), then \(\mu (H+x) = 0\) or 1, \(x \in R^ \infty\), for each \(\mu\)-measurable subgroup \(H\) of \(R^ \infty\).
scientific article; zbMATH DE number 599755