  Mutual information is one of the essential building blocks of information
theory. Yet, it is only finitely defined for distributions with fast decaying
tails on a countable joint alphabet of two random elements. The unboundedness
of mutual information over the general class of all distributions on a joint
alphabet prevents its potential utility to be fully realized. This is in fact a
void in the foundation of information theory that needs to be filled. This
article proposes a family of generalized mutual information all of whose
members 1) are finitely defined for each and every distribution of two random
elements on a joint countable alphabet, except the one by Shannon, and 2) enjoy
all utilities of a finite Shannon's mutual information.
