Exact conditions for countable inclusion-exclusion identity and   extensions
  We give simple necessary and sufficient conditions for the inclusion-exclusion identity to hold for an infinite countable number of sets. In terms of a random variable, whose range are nonnegative integers, this condition is equivalent to the convergence to zero of binomial moments. Some standard extensions of the countable inclusion-exclusion identity are also given. 
Shmuel Friedland and Elliot Krop