  In studies of the quantum Zeno and anti-Zeno effects, it is usual to consider
rapid projective measurements with equal time intervals being performed on the
system to check whether or not the system is in the initial state. These
projective measurements are selective measurements in the sense that the
measurement results are read out and only the case where all the measurement
results correspond to the initial state is considered in the analysis of the
effect of the measurements. In this paper, we extend such a treatment to
consider the effect of repeated non-selective measurements - only the final
measurement is required to correspond to the initial state, while we do not
know the results of the intermediate measurements. We present a general
formalism to derive the effective decay rate of the initial quantum state with
such non-selective measurements. Importantly, we show that there is a
difference between using non-selective measurements and the usual approach of
considering only selective measurements only if we go beyond the weak
system-environment coupling regime in models other than the usual population
decay models. As such, we then apply our formalism to investigate the quantum
Zeno and anti-Zeno effects for three exactly solvable system-environment
models: a single two-level system undergoing dephasing, a single two-level
system interacting with an environment of two-level systems, and a large spin
undergoing dephasing. Our results show that the quantum Zeno and anti-Zeno
effects in the presence of non-selective measurements can differ very
significantly as compared to the repeated selective measurement scenario.
