Consistency among prior, posteriors, and information sets implies information sets form a partition almost surely, posteriors equal Bayes updates, and qualitative beliefs become fully introspective knowledge in standard settings.
Halpern (1996): Should Knowledge Entail Belief? Journal of Philosphical Logic 25, pp
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On the Consistency among Prior, Posteriors, and Information Sets (Extended Abstract)
Consistency among prior, posteriors, and information sets implies information sets form a partition almost surely, posteriors equal Bayes updates, and qualitative beliefs become fully introspective knowledge in standard settings.