Compact convergence of σ-fields and relaxed conditional expectation

• 成果类型: Article
• 署名作者: Artstein, Z
• 署名单位: Weizmann Institute of Science; Universite de Montpellier
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/PL00008787
• 发表日期: 2001
• 页码: 369-394
• 关键词: information; continuity

摘要

The collection of sub-sigma -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 sigma -fields. A notion of relaxed conditional expectation with respect to it relaxed sigma -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.