A compact topology for $\sigma$-algebra convergence
Probability
2021-05-20 v3 Functional Analysis
Abstract
We propose a sequential topology on the space of sub--algebras of a separable probability space by linking conditional expectations on along sequences of sub--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--algebras. The proposed topology preserves independence and is compatible with join and meet operations.
Cite
@article{arxiv.1802.05920,
title = {A compact topology for $\sigma$-algebra convergence},
author = {Patrick Beissner and Jonas M. Tölle},
journal= {arXiv preprint arXiv:1802.05920},
year = {2021}
}
Comments
There 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