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Published March 23, 2016 | Accepted Version
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Measurability Is Not About Information

Abstract

We comment on the relation between models of information based on signals/partitions, and those based on σ-algebras. We show that more informative signals need not generate finer σ-algebras, hence that Blackwell's theorem fails if information is modeled as σ-algebras. The reason is that the σ-algebra generated by a partition does not contain all the events that can be known from the information provided by the signal. We also show that there is a non-conventional σ-algebra that can be associated to a signal which does preserve its information content. Further, expectations and conditional expectations may depend on the choice of σ-algebra that is associated to a signal. We provide a simple characterization of when the model is robust to changes in the σ-algebras.

Additional Information

March 19, 2001. We are grateful to David Blackwell, Don Brown, David Levine, Jim Pitman, Maxwell Stinchcombe, Tarun Sabarwal, Bill Sudderth, Sergio Turner, Bill Zame, and seminar participants at UCLA and Yale for comments. We thank Catherine Huafei Yan, Ernesto Mordecki, Malempati Rao, and Srinivasa Varadhan for their kind attention to our questions.

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August 19, 2023
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January 13, 2024