  The manifestation of measurements, randomly distributed in time, on the
evolution of quantum systems are analyzed in detail. The set of randomly
distributed measurements (RDM) is modeled within the renewal theory, in which
the distribution is characterized by the probability density function (PDF)
W(t) of times t between successive events (measurements). The evolution of the
quantum system affected by the RDM is shown to be described by the density
matrix satisfying the stochastic Liouville equation. This equation is applied
to the analysis of the RDM effect on the evolution of a two level systems for
different types of RDM statistics, corresponding to different PDFs W(t).
Obtained general results are illustrated as applied to the cases of the
Poissonian [W(t) ~ e^{-w_r t}] and anomalous [W(t) ~ 1/t^{1+\alpha}, (\alpha is
smaller or equal to 1)], RDM statistics. In particular, specific features of
the quantum and inverse Zeno effects, resulting from the RDM, are thoroughly
discussed.
