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.

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