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Compact convergence of σ-fields and relaxed conditional expectation
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  • Published: July 2001

Compact convergence of σ-fields and relaxed conditional expectation

  • Zvi Artstein1 

Probability Theory and Related Fields volume 120, pages 369–394 (2001)Cite this article

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Abstract.

The collection of sub-σ-fields of a Borel measure space when endowed with the topology of strong convergence is in general not a compact space. The paper offers a completion of this space which makes it compact. The elements which are added to the space are called relaxed σ-fields. A notion of relaxed conditional expectation with respect to a relaxed σ-field is identified. The relaxed conditional expectation is a probability measure-valued map. It is shown that the conditional expectation operator is continuous on the completion of the space. Other properties of conditional expectation are lifted to and interpreted in the relaxed framework.

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Authors and Affiliations

  1. Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. e-mail: zvika@wisdom.weizmann.ac.il, , , , , , IL

    Zvi Artstein

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  1. Zvi Artstein
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Received: 22 February 1999 / Revised version: 23 October 2000 / Published online: 8 May 2001

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Artstein, Z. Compact convergence of σ-fields and relaxed conditional expectation. Probab Theory Relat Fields 120, 369–394 (2001). https://doi.org/10.1007/PL00008787

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  • Issue Date: July 2001

  • DOI: https://doi.org/10.1007/PL00008787

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  • Mathematics Subject Classification (2000): 28A05, 28A33, 60A10, 60B10
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