Scales
Echelles
Résumé
We introduce the notion of scale to generalize some aspects of dimension theory. Several versions of scales exist such as Hausdorff, packing, box, local and quantization. We prove general theorems comparing them. They are applied to describe the largeness of ergodic decompositions, Wiener measure and functional spaces. The first application solves a problem set by Berger on the notions of emergence (2020); the second lies in the study of small ball estimates of the Wiener measure and extends an estimation of Dereich-Lifshits (2005); the last refines the Kolmogorov-Tikhomirov theorems (1958) on the ϵ-capacity of functional spaces.
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