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On Strongly Equivalent Nonrandomized Transition Probabilities

Dvoretzky, Wald, and Wolfowitz proved in 1951 the existence of equivalent and strongly equivalent mappings for a given transition probability when the number of nonatomic measures is finite and the decision set is finite. This paper introduces a notion of strongly equivalent transition probabilities with respect to a finite collection of functions. This notion contains the notions of equivalent and strongly equivalent transition probabilities as particular cases. This paper shows that a strongly equivalent mapping with respect to a finite collection of functions exists for a finite number of nonatomic distributions and finite decision set. It also provides a condition when this is true for a countable decision set. According to a recent example by Loeb and Sun, a strongly equivalent mapping may not exist under these conditions when the decision set is uncountable. This paper also provides two additional counterexamples and shows that strongly equivalent mappings exist for homogeneous transition probabilities.