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 gG is called an admissible (partially admissible, singular, equivalent, invariant) transformation for μ relative to u if μ_gu (accordingly: μ_g otu, μ_gu, μ_gu, μ_g=c· u), where μ_g(E):=μ(g^-1E). We denote its collection by A(μ\!\!\!\! u) (accordingly: AP(μ\!\!\!\! u), S(μ\!\!\!\! u), E(μ\!\!\!\! u), I(μ\!\!\!\! u)). It is shown that all these sets are Borel subsets of very bounded types. In particular, A(μ\!\!\!\! u) is a G_δσδ-subset of G. If G is a Polish group, then A(μ\!\!\!\! u), E(μ\!\!\!\! u) and I(μ\!\!\!\! u) admit a Polish topology. Mathematics Subject Classification: 28C99, 37A99.