We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda~Jr. We show that these two notions coincide, if the measure is doubling and supports a $1$-Poincar\'e inequality. In doing so, we also prove that if the measure is doubling and supports a $1$-Poincar\'e inequality, then the metric space supports a \emph{Semmes family of curves} structure.

中文翻译：

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度量空间中两个 BV 类函数的等价性，以及 1-庞加莱不等式下 Semmes 曲线族的存在

我们考虑完全度量空间中的有界变差函数的两个概念，一个是由于 Martio，另一个是由于 Miranda~Jr。我们证明这两个概念是一致的，如果该度量加倍并且支持 $1$-Poincar\'e 不等式。这样做，我们还证明了如果度量加倍并支持 $1$-Poincar\'e 不等式，那么度量空间支持 \emph{Semmes 曲线族}结构。