完备连通一维度量空间上的有界变差函数
Functions of bounded variation on complete and connected one-dimensional metric spaces
摘要：
本文研究了完备连通度量空间上具有有限一维Hausdorff测度的有界变差函数。基于逐点变化的紧区间上的BV函数的定义被扩展到这个一般设置。我们证明了这种BV函数的定义等价于Miranda引入的BV函数。此外，我们还研究了Federer对度量测度空间上有限周长集的刻画中底层空间条件的必要性。特别是，我们的例子表明，加倍和庞加莱不等式条件是必不可少的，表明一个集有有限的周长，如果余维1豪斯多夫测度的测度理论的边界是有限的。
英文摘要：
In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda. Furthermore, we study the necessity of conditions on the underlying space in Federer's characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.
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