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A metric space associated with a probability space. (English) Zbl 0770.28001
Summary: For a complete probability space \((\Omega,\Sigma,P)\), the set of all complete sub-\(\sigma\)-algebras of \(\Sigma\), \(S(\Sigma)\), is given a natural metric and studied. The questions of when \(S(\Sigma)\) is compact or connected are answered and the important subset consisting of all continuous sub-\(\sigma\)-algebras is shown to be closed. Connections with Christensen’s metric on the von Neumann subalgebras of a Type \(\text{II}_ 1\)-factor are briefly discussed.

MSC:
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
46L10 General theory of von Neumann algebras
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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