On continuous Polish group actions and equivalence relations

Nikolaos E. Sofronidis

Published 2014-08-09, updated 2015-02-26Version 2

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 ${\bf P}$.