A metric space associated with a probability space (Q1210452)
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scientific article; zbMATH DE number 179245
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A metric space associated with a probability space |
scientific article; zbMATH DE number 179245 |
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A metric space associated with a probability space (English)
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8 August 1993
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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.
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\(\sigma\)-algebras
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conditional expectations
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metric space
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complete probability space
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complete sub-\(\sigma\)-algebras
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Christensen's metric
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von Neumann subalgebras
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Type \(\text{II}_ 1\)-factor
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