A couple of remarks on the convergence of $\sigma$-fields on probability spaces

TL;DR

This paper reviews various modes of convergence of sub-$\sigma$-fields on probability spaces, highlighting their properties such as preservation of independence and invariance under measure changes, and discusses partial results for operator-norm convergence.

Contribution

It provides a comparative analysis of different convergence modes of sub-$\sigma$-fields and notes their key properties and partial results for operator-norm convergence.

Findings

All convergence modes preserve independence.

All are invariant under equivalent measure changes.

Partial results are obtained for operator-norm convergence.

Abstract

The following modes of convergence of sub-$\sigma$-fields on a given probability space have been studied in the literature: weak convergence, strong convergence, convergence with respect to the Hausdorff metric, almost-sure convergence, set-theoretic convergence, monotone convergence. It is noted that all preserve independence, and all are invariant under passage to an equivalent probability measure. Partial results for the case of operator-norm convergence obtain.