On continuous Polish group actions and equivalence relations

Nikolaos E. Sofronidis

paper

On continuous Polish group actions and equivalence relations

arXiv:1408.2097

Abstract

Let $X = {P \in [0, 1]^{N} : (\forall ν \in N) (P ({ν}) > 0) \land ν = 0 \sum \infty P ({ν}) = 1}$ be the Polish space of probability measures on $N$ , each of which assigns positive probability to every elementary event, while for any $P \in X$ , let $Γ_{P} = {ξ \in L^{1} (N, P) : (\forall ν \in N) (ξ (ν) > 0) \land ν = 0 \sum \infty ξ (ν) P ({ν}) = 1}$ and let $Φ_{P} : Γ_{P} ∋ ξ \mapsto Φ_{P} (ξ) \in X$ be defined by the relation $(Φ_{P} (ξ)) ({ν}) = ξ (ν) P ({ν})$ , whenever $ν \in N$ . If we consider the equivalence relation $E = {(P, Q) \in X^{2} : (\exists ξ \in Γ_{P}) (Q = Φ_{P} (ξ))}$ , the Polish space $P = {x \in ℓ^{1} (R) : (\forall n \in N) (x (n) > 0)}$ and the commutative Polish group $G = {g \in (0, \infty)^{N} : n \to \infty lim g (n) = 1}$ , while we set $(g \cdot x) (n) = g (n) x (n)$ , whenever $g \in G$ , $x \in P$ and $n \in N$ , then $E$ is definable and it admits a strong approximation by the turbulent Polish group action of $G$ on $P$ .