Kudo-Continuity Of Entropy Functionals

TL;DR

This paper introduces Kudo-continuity for entropy functionals on sub-$\sigma$-algebras, demonstrating many are Kudo-continuous and exploring their continuity properties, with applications to entropy spectra of group boundaries.

Contribution

It defines Kudo-continuity for entropy functionals and proves many entropy measures possess this property, advancing understanding of their stability and convergence behaviors.

Findings

Many entropy functionals are Kudo-continuous.

Established upper and lower continuity under stochastic domination.

Applied results to entropy spectra of group boundaries.

Abstract

We study in this paper real-valued functions on the space of all sub-$\sigma$-algebras of a probability measure space, and introduce the notion of Kudo-continuity, which is an a priori strengthening of continuity with respect to strong convergence. We show that a large class of entropy functionals are Kudo-continuous. On the way, we establish upper and lower continuity of various entropy functions with respect to asymptotic second order stochastic domination, which should be of independent interest. An application to the study of entropy spectra of $\mu$-boundaries associated to random walks on locally compact groups is given.