A metric space associated with probability space

Abstract

For a complete probability space (Ω,∑,P), the set of all complete sub-σ-algebras of ∑, S(∑), is given a natural metric and studied. The questions of when S(∑) is compact or connected are awswered and the important subset consisting of all continuous sub-σ-algebras is shown to be closed. Connections with Christensen's metric on the von Neumann subalgebras of a Type II1-factor are briefly discussed.

Keywords

• σ-algebras
• conditional expectations
• metric space
• von Neumann algebra.