  We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure
preserving dynamical system $(X, \mu, T)$ with a compatible metric $d$. We
prove that, under some regularity conditions, the $\mu$-measure of the
following set \[
  R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\N
\} \] obeys a zero-full law according to the convergence or divergence of a
certain series, where $\psi:\N\to\R^+$. Some of the applications of our main
theorem include the continued fractions dynamical systems, the beta dynamical
systems, and the homogeneous self-similar sets.
