Zero probabilities in stochastic independence

In a previous paper we pointed out the lack of the classic definition of stochastic independence: a stronger one (that avoids the main inconsistencies related to events of probability 0 or 1) has been introduced in that paper. We give here a new “improved” definition of independence of two events (coinciding with the former one when some relevant events have probabilities different from 0 and 1): it allows to deepen and clarify further aspects (in particular those connected with the possible “symmetry” of this important probabilistic concept) and to establish a characterization theorem.