Normalization and φ-function: Definition and Consequences.

It is known from the literature that a phi-function may be used to construct the phi-families of probability distributions. In this paper, we assume that one of the properties in the definition of phi-function is not satisfied and we analyze the behavior of the normalizing function near the boundary of its domain. As a consequence, we find a measurable function that does not belong to the Musielak-Orlicz class, but the normalizing function applied to this found function converges to a finite value near the boundary of its domain. We conclude showing that this change in the definition of phi-function affects the behavior of the normalizing function.