Topological Properties of the Set of Admissible Transformations of Measures - Gabriyelyan, S. S. (Журнал математичної фізики, аналізу, геометрії, 2006)
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 $μ$ relative to $ u$ if $μ_g\ll u$ (accordingly: $μ_g ot\perp u$, $μ_g\perp u$, $μ_g\sim u$, $μ_g=c· u$), where $μ_g(E):=μ(g^{-1}E)$. We denote its collection by $A(μ\!\!\mid\!\! u)$ (accordingly: $AP(μ\!\!\mid\!\! u)$, $S(μ\!\!\mid\!\! u)$, $E(μ\!\!\mid\!\! u)$, $I(μ\!\!\mid\!\! u)$). It is shown that all these sets are Borel subsets of very bounded types. In particular, $A(μ\!\!\mid\!\! u)$ is a $G_{δσδ}$-subset of $G$. If $G$ is a Polish group, then $A(μ\!\!\mid\!\! u)$, $E(μ\!\!\mid\!\! u)$ and $I(μ\!\!\mid\!\! u)$ admit a Polish topology. Mathematics Subject Classification: 28C99, 37A99.