On the Difference Between the Cardinalities of Measurable Value Functions and Von Neumann-Morgenstern Utility Functions

Dyer and Sarin (Dyer, J. S., R. K. Sarin. 1979. Measurable multiattribute value functions. Opns. Res.27 810–822.) proposed an important generalization of the additive-multiplicative decomposition theorem for von Neumann-Morgenstern (vNM) utility functions to measurable value functions based on the cardinality property of measurable value functions. However, because of a difference between the cardinality of measurable value functions and the cardinality of vNM utility functions, an additional technical condition is needed, and presented here, for the decomposition theorem to hold.