A compact topology for σ-algebra convergence

2018/02/16 by Beissner, Patrick, Tölle, Jonas M.
#28A05 #55R65 (Secondary) #60A10 #60B05 (Primary) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1802.05920

Abstract

We propose a sequential topology on the space of sub-σ-algebras of a separable probability space (Ω,F,ℙ) by linking conditional expectations on L² along sequences of sub-σ-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-σ-algebras. The proposed topology preserves independence and is compatible with join and meet operations.

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