Admissible transformations of measures - Gabriyelyan, S. S. (Журнал математичної фізики, аналізу, геометрії, 2005)
Let a topological semigroup $G$ acts on a topological space $X$. A transformation $g\in G$ is called an admissible (partially admissible, singular, equivalent, invariant) transform 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: $AP(μ\!\!\mid\!\! u)$, $S(μ\!\!\mid\!\! u)$, $E(μ\!\!\mid\!\! u)$, $I(μ\!\!\mid\!\! u)$). The algebraic and the measure theoretical properties of these sets are studied. It is done the Lebesgue-type decomposition. If $G=X$ is a locally compact group, we give some informations about the measure theoretical size of $A(μ)$. Mathematical Subject Classification: 28C99, 37A99.