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.

arXiv commentThere is a gap in the proof of Theorem 2.6 on page 12, Proof of Claim 3: the tower-type property cannot be applied in the asserted generality

Cited by in corpus (1)

• A potpourri of results in the general theory of stochastic noises