On the difference between the cardinalities of measurable value functions and von Neumann–Morgenstern utility functions

Dyer and Sarin 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.