Admissible transformations of measures

S. S. Gabrielyan

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 1 (2005), pp. 155-181

Résumé

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 $\mu$ 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 $A(\mu|\nu)$ (accordingly: $AP(\mu|\nu)$, $S(\mu|\nu)$, $E(\mu|\nu)$, $I(\mu|\nu)$). 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 $A(\mu)$.