A compact topology for $\sigma$-algebra convergence

TL;DR

This paper introduces a new compact topology on the space of sub-$\sigma$-algebras of a separable probability space, using conditional expectations and bundle space construction, preserving independence and lattice operations.

Contribution

It develops a sequential topology on sub-$\sigma$-algebras that is compact, preserves independence, and aligns with join and meet operations, advancing the understanding of $\sigma$-algebra convergence.

Findings

Established the compactness of the space of sub-$\sigma$-algebras.

Proposed a topology that preserves independence.

Ensured compatibility with join and meet operations.

Abstract

We propose a sequential topology on the space of sub-$\sigma$-algebras of a separable probability space $(\Omega,\mathcal{F},\mathbb{P})$ by linking conditional expectations on $L^{2}$ along sequences of sub-$\sigma$-algebras. The varying index of measurability is captured by a bundle space construction. As a consequence, we establish the compactness of the space of sub-$\sigma$-algebras. The proposed topology preserves independence and is compatible with join and meet operations.