Uniqueness of a Furstenberg system

Given a countable amenable group $G$, a Følner sequence $(F_N) \subseteq G$, and a set $E \subseteq G$ with $\bar{d}_{(F_N)}(E)=\limsup_{N \to \infty} \frac{|E \cap F_N|}{|F_N|}>0$, Furstenberg's correspondence principle associates with the pair $(E,(F_N))$ a measure preserving system $(X,\mathcal{B},μ,(T_g)_{g \in G})$ and a set $A \in \mathcal{B}$ with $μ(A)=\bar{d}_{(F_N)}(E)$, in such a way that for all $r \in \mathbb{N}$ and all $g_1,\dots,g_r \in G$ one has $\bar{d}_{(F_N)}(g_1^{-1}E \cap \dots \cap g_r^{-1}E)\geqμ((T_{g_1})^{-1}A \cap \dots \cap (T_{g_r})^{-1}A)$. We show that under some natural assumptions, the system $(X,\mathcal{B},μ,(T_g)_{g \in G})$ is unique up to a measurable isomorphism. We also establish variants of this uniqueness result for non-countable discrete amenable semigroups as well as for a generalized correspondence principle which deals with a finite family of bounded functions $f_1,\dots,f_{\ell}: G \rightarrow \mathbb{C}$.

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