We consider vector valued mappings defined on metric measure spaces with a measurable differentiable structure and study both approximations by nicer mappings and regular extensions of the given mappings when defined on closed subsets. Therefore, we propose a first approach to these problems, largely studied on Euclidean and Banach spaces during the last century, for first order differentiable functions de-fined on these metric measure spaces.

中文翻译：

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度量测度空间上可微函数的密度和扩展

我们考虑在具有可测量的可微结构的度量度量空间上定义的向量值映射，并通过更好的映射和在封闭子集上定义的给定映射的常规扩展来研究近似值。因此，我们提出了解决这些问题的第一种方法，在上个世纪主要研究了欧几里得空间和巴拿赫空间，用于在这些度量测度空间上定义的一阶可微函数。