Uniform measures on the arbitrary compact metric spaces, with applications

TL;DR

This paper introduces the concept of uniform measures on arbitrary compact metric spaces and explores their applications in embedding theorems and random process theory, particularly in majorizing measure methods.

Contribution

It proposes a new notion of uniform measure on compact metric spaces and applies it to embedding theorems and the analysis of random fields using majorizing measure techniques.

Findings

Defined uniform measures on compact metric spaces.

Connected uniform measures to embedding theorems.

Applied to the theory of random processes and majorizing measures.

Abstract

We introduce and investigate in this short report the new notion of uniform measure (distribution) on the arbitrary compact metric space. We consider also some possible applications of these measures in the theory of imbedding theorems and in the theory of random processes (fields), in particular, in the so-called majorizing (and minorizing) measures method, belonging to X.Fernique and M.Talagrand. These considerations based on the L.Arnold and P.Imkeller generalization of the classical A.M.Garsia-E.Rodemich-H.Jr.Rumsey inequality and X.Fernique-M.Talagrand estimation for random fields.