  In the study of some dynamical systems the limsup set of a sequence of
measurable sets is often of interest. The shrinking targets and recurrence are
two of the most commonly studied problems that concern limsup sets. However,
the zero-one laws for the shrinking targets and recurrence are usually treated
separately and proved differently. In this paper, we introduce a generalized
definition that can specialize into the shrinking targets and recurrence; our
approach gives a unified proof of the zero-one laws for the two problems.
