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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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