Scales

TL;DR

This paper introduces the concept of scales to compare various invariants of metric spaces and measures, providing a unified framework that enhances understanding of infinite-dimensional geometries and solves existing problems.

Contribution

It develops multiple versions of scales for metric spaces, establishes comparison theorems, and applies these to ergodic decompositions, Wiener measure geometry, and functional space analysis.

Findings

Solved Berger's emergence problem using scales.

Extended Dereich-Lifshits' work on Wiener measure.

Refined Kolmogorov-Tikhomirov's analysis of functional spaces.

Abstract

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.