Topological Properties of the Set of Admissible Transformations of Measures

Анотація

Suppose a topological semigroup $G$ acts on a topological space $X$. A transformation $g\in G$ is called an admissible (partially admissible, singular, equivalent, invariant) transformation for $\mu$ relative to $\nu$ if $\mu_g\ll\nu$ (accordingly: $\mu_g\not\perp\nu$, $\mu_g\perp\nu$, $\mu_g\sim\nu$, $\mu_g=c\cdot \nu$), where $\mu_g(E):=\mu(g^{-1}E)$. We denote its collection by $A(\mu\!\!\mid\!\!\nu)$ (accordingly: $AP(\mu\!\!\mid\!\!\nu)$, $S(\mu\!\!\mid\!\!\nu)$, $E(\mu\!\!\mid\!\!\nu)$, $I(\mu\!\!\mid\!\!\nu)$). It is shown that all these sets are Borel subsets of very bounded types. In particular, $A(\mu\!\!\mid\!\!\nu)$ is a $G_{\delta\sigma\delta}$-subset of $G$. If $G$ is a Polish group, then $A(\mu\!\!\mid\!\!\nu)$, $E(\mu\!\!\mid\!\!\nu)$ and $I(\mu\!\!\mid\!\!\nu)$ admit a Polish topology.

Mathematics Subject Classification: 28C99, 37A99.