论文标题
秤
Scales
论文作者
论文摘要
我们介绍了规模的概念，以概括和比较不同的度量空间及其措施的不同不变。引入了几种版本的量表，例如Hausdorff，包装，框，本地和量化。此外，它们的定义是不同的生长，尤其是对无限尺寸空间的精致研究。我们证明了比较量表不同版本的一般定理。它们被应用于描述千古分解，维纳措施和功能空间的几何形状。第一个申请解决了伯格在出现概念上的问题（2020）；第二个在于Wiener措施的几何形状，并扩展了Dereich-Lifshits的工作（2005年）。最后一项完善了Kolmogorov-Tikhomirov（1958）关于功能空间的研究。
We introduce the notion of scale to generalize and compare different invariants of metric spaces and their measures. Several versions of scales are introduced such as Hausdorff, packing, box, local and quantization. They moreover are defined for different growth, allowing in particular a refined study of infinite dimensional spaces. We prove general theorems comparing the different versions of scales. They are applied to describe geometries of ergodic decompositions, of the Wiener measure and of functional spaces. The first application solves a problem of Berger on the notions of emergence (2020); the second lies in the geometry of the Wiener measure and extends the work of Dereich-Lifshits (2005); the last refines Kolmogorov-Tikhomirov (1958) study on functional spaces.
