Standardness as an invariant formulation of independence

TL;DR

This paper explores the concept of standardness in filtrations, providing detailed definitions, characterizations, and introducing new notions related to nonstandard filtrations, with implications across probability, combinatorics, and algebra.

Contribution

It offers a comprehensive analysis of Markov standard filtrations and introduces the concept of shadow metric-measure spaces for nonstandard filtrations.

Findings

Characterization of Markov standard filtrations

Standardness criterion as a key condition

Introduction of shadow metric-measure spaces

Abstract

The notion of a homogeneous standard filtration of $\sigma$-algebras was introduced by the author in 1970. The main theorem asserted that a homogeneous filtration is standard, i.e., generated by a sequence of independent random variables, if and only if the standardness criterion is satisfied. In this paper we give detailed definitions and characterizations of Markov standard filtrations. The notion of standardness is essential for applications of probabilistic, combinatorial, and algebraic nature. At the end of the paper we present new notions of {\it shadow metric-measure space} related to nonstandard filtrations.