  The law of large numbers is one of the most fundamental results in
Probability Theory. In the case of independent sequences, there are some known
characterizations; for instance, in the independent and identically distributed
setting it is known that the law of large numbers is equivalent to
integrability. In the case of dependent sequences, there are no known general
characterizations -- to the best of our knowledge. We provide such a
characterization for identically distributed Bernoulli sequences in terms of a
product disintegration.
