A domain-theoretic investigation of posets of sub-sigma-algebras (extended abstract)

TL;DR

This paper explores the domain-theoretic properties of posets of sub-sigma-algebras in measurable spaces, revealing limitations and counterexamples that challenge previous assumptions and connections with stochastic relations.

Contribution

It provides a domain-theoretic analysis showing that posets of sub-sigma-algebras lack certain desirable properties, offering counterexamples relevant to stochastic relation congruences.

Findings

Posets of sub-sigma-algebras do not have certain domain-theoretic properties.

Counterexamples apply to smooth equivalence relations on analytic spaces.

Challenges previous assumptions about lattice properties in this context.

Abstract

Given a measurable space (X, M) there is a (Galois) connection between sub-sigma-algebras of M and equivalence relations on X. On the other hand equivalence relations on X are closely related to congruences on stochastic relations. In recent work, Doberkat has examined lattice properties of posets of congruences on a stochastic relation and motivated a domain-theoretic investigation of these ordered sets. Here we show that the posets of sub-sigma-algebras of a measurable space do not enjoy desired domain-theoretic properties and that our counterexamples can be applied to the set of smooth equivalence relations on an analytic space, thus giving a rather unsatisfactory answer to Doberkat's question.