On continuous Polish group actions and equivalence relations

TL;DR

This paper investigates the structure of a specific equivalence relation on probability measures on natural numbers, showing it can be strongly approximated by a turbulent group action on a related Polish space.

Contribution

It establishes that the equivalence relation defined by measure transformations is analyzable via a turbulent Polish group action, providing new insights into the dynamics of such relations.

Findings

The equivalence relation is definable within the given Polish space.

It admits a strong approximation by a turbulent Polish group action.

The group action is shown to be continuous and turbulent.

Abstract

Let $X = \left\{P \in [0,1]^{\bf N} : \left(\forall \nu \in {\bf N} \right) \left(P \left(\{\nu \} \right) > 0 \right) \wedge \sum\limits_{\nu = 0}^{\infty} P \left(\{\nu \} \right) = 1 \right\} $ be the Polish space of probability measures on ${\bf N}$, each of which assigns positive probability to every elementary event, while for any $P \in X$, let ${\Gamma}_{P} = \left\{\xi \in L^{1}({\bf N}, P) : \left(\forall \nu \in {\bf N} \right) \left(\xi (\nu) > 0 \right) \wedge \sum\limits_{\nu = 0}^{\infty} \xi (\nu) P \left(\{\nu \} \right) = 1 \right\} $ and let ${\Phi}_{P} : {\Gamma}_{P} \ni \xi \mapsto {\Phi}_{P}(\xi) \in X$ be defined by the relation $\left({\Phi}_{P}(\xi) \right) \left(\{\nu \} \right) = \xi (\nu) P \left(\{\nu \} \right) $, whenever $\nu \in {\bf N}$. If we consider the equivalence relation $E = \left\{(P,Q) \in X^{2} : \left(\exists \xi \in {\Gamma}_{P} \right) \left(Q = {\Phi}_{P}(\xi) \right) \right\} $, the Polish space ${\bf P} = \left\{{\bf x} \in {\ell}^{1} \left({\bf R} \right) : \left(\forall n \in {\bf N} \right) \left({\bf x}(n) > 0 \right) \right\} $ and the commutative Polish group ${\bf G} = \left\{{\bf g} \in (0, \infty)^{\bf N} : \lim\limits_{n \rightarrow \infty}{\bf g}(n) = 1 \right\} $, while we set $\left({\bf g} \cdot {\bf x} \right) (n) = {\bf g}(n){\bf x}(n)$, whenever ${\bf g} \in {\bf G}$, ${\bf x} \in {\bf P}$ and $n \in {\bf N}$, then $E$ is definable and it admits a strong approximation by the turbulent Polish group action of ${\bf G}$ on ${\bf P}$.