论文标题
对应于数字的驯服集上的严格单调度量
A strictly monotone measure on tame sets that corresponds to a numerosity
论文作者
论文摘要
从几何测量理论中适应标准方法，我们提供了一个多项式值测量$μ$的示例，$ \ mathbb {r}^d $中的温和套件满足了许多理想的属性。其中是严格的单调性：适当的子集的度量严格少于整个集合的量度。使用非标准分析中的技术，我们显示$μ$的域可以扩展到$ \ Mathbb {r}^d $的所有子集（最多达到等价模量InfitInitesimals）。最终的扩展是一个数字函数，该函数编码了所有$ i \ in \ mathbb {n} $的$ i $ dimensional hausdorff度量，以及$ i $ th的内在卷函数。
Adapting standard methods from geometric measure theory, we provide an example of a polynomial-valued measure $μ$ on tame sets in $\mathbb{R}^d$ which satisfies many desirable properties. Among these is strict monotonicity: the measure of a proper subset is strictly less than the measure of the whole set. Using techniques from non-standard analysis, we display that the domain of $μ$ can be extended to all subsets of $\mathbb{R}^d$ (up to equivalence modulo infinitesimals). The resulting extension is a numerosity function that encodes the $i$-dimensional Hausdorff measure for all $i\in \mathbb{N}$, as well as the $i$-th intrinsic volume functions.
