The Topology of Information on the Space of Probability Measures over Polish Spaces

TL;DR

This paper investigates the topology of information on probability measures over Polish spaces, demonstrating that convergence preserves conditional independence properties, and corrects a previous proof in the literature.

Contribution

It provides a rigorous proof that the topology of information preserves conditional independence under convergence and rectifies an earlier lemma's proof.

Findings

Convergence in the topology of information preserves conditional independence.

Provides a corrected proof of a key lemma from prior work.

Enhances understanding of measure convergence in Polish spaces.

Abstract

We study here the topology of information on the space of probability measures over Polish spaces that was defined in [1]. We show that under this topology, a convergent sequence of probability measures satisfying a conditional independence property converges to a measure that also satisfies the same conditional independence property. In the process of showing this, we also provide a correct proof of the result in Lemma 4 in [1].