Borel reducibility and finitely Holder(α) embeddability

Let $(X_n,d_n),\,n\in\Bbb N$ be a sequence of pseudo-metric spaces, $p\ge 1$. For $x,y\in\prod_{n\in\Bbb N}X_n$, let $(x,y)\in E((X_n)_{n\in\Bbb N};p)\Leftrightarrow\sum_{n\in\Bbb N}d_n(x(n),y(n))^p<+\infty$. For Borel reducibility between equivalence relations $E((X_n)_{n\in\Bbb N};p)$, we show it is closely related to finitely Hölder($α$) embeddability between pseudo-metric spaces.

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