Ranks of Maharam algebras

TL;DR

This paper constructs examples of Maharam algebras with arbitrarily high exhaustivity ranks using Schreier families, demonstrating the diversity of such algebras beyond measure algebras.

Contribution

It introduces a method to generate exhaustive submeasures of any high rank, leading to uncountably many non-isomorphic separable atomless Maharam algebras.

Findings

Existence of exhaustive submeasures with arbitrary high exhaustivity rank

Construction of uncountably many non-isomorphic Maharam algebras

Use of Schreier families and norms for construction

Abstract

Solving a well-known problem of Maharam, Talagrand [17] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure one can canonically assign a certain countable ordinal, its exhaustivity rank. In this paper, we use carefully constructed Schreier families and norms derived from them to provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This gives rise to uncountably many non isomorphic separable atomless Maharam algebras.