A trichotomy for a class of equivalence relations

Let $X_n, n\in\Bbb N$ be a sequence of non-empty sets, $ψ_n:X_n^2\to\Bbb R^+$. We consider the relation $E((X_n,ψ_n)_{n\in\Bbb N})$ on $\prod_{n\in\Bbb N}X_n$ by $(x,y)\in E((X_n,ψ_n)_{n\in\Bbb N})\Leftrightarrow\sum_{n\in\Bbb N}ψ_n(x(n),y(n))<+\infty$. If $E((X_n,ψ_n)_{n\in\Bbb N})$ is a Borel equivalence relation, we show a trichotomy that either $\Bbb R^\Bbb N/\ell_1\le_B E$, $E_1\le_B E$, or $E\le_B E_0$. We also prove that, for a rather general case, $E((X_n,ψ_n)_{n\in\Bbb N})$ is an equivalence relation iff it is an $\ell_p$-like equivalence relation.

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