A compact topology for $\sigma$-algebra convergence

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.