  In a previous paper: A. Paszkiewicz, T. Sobieszek, Additive Entropies of
Partitions, we have given a description of additive partition entropies that is
real functions $I$ on the set of finite partitions that are additive on
stochastically independent partitions in a given probability space. We now
present an analogical result, this time without assuming continuity.
  As a by-product of our efforts we solve a 2-cocycle functional equation for
certain subsets of convex cones.
