  In this short paper, we consider a quadruple $(\Omega, \AA, \theta,
\mu)$,where $\AA$ is a $\sigma$-algebra of subsets of $\Omega$, and $\theta$ is
a measurable bijection from $\Omega$ into itself that preserves the measure
$\mu$. For each $B \in \AA$, we consider the measure $\mu_B$ obtained by taking
cycles (excursions) of iterates of $\theta$ from $B$. We then derive a relation
for $\mu_B$ that involves the forward and backward hitting times of $B$ by the
trajectory $(\theta^n \omega, n \in \Z)$ at a point $\omega \in \Omega$.
Although classical in appearance, its use in obtaining uniqueness of invariant
measures of various stochastic models seems to be new. We apply the concept to
countable Markov chains and Harris processes.
