  We address the statistics of continuous weak linear measurement on a
few-state quantum system that is subject to a conditioned quantum evolution.
For a conditioned evolution, both the initial and final states of the system
are fixed: the latter is achieved by the post-selection in the end of the
evolution. The statistics may drastically differ from the non-conditioned case,
and the interference between initial and final states can be observed in the
probability distributions of measurement outcomes as well as in the average
values exceeding the conventional range of non-conditioned averages. We develop
a proper formalism to compute the distributions of measurement outcomes,
evaluate and discuss the distributions in experimentally relevant setups. We
demonstrate the manifestations of the interference between initial and final
states in various regimes. We consider analytically simple examples of
non-trivial probability distributions. We reveal peaks (or dips) at
half-quantized values of the measurement outputs. We discuss in detail the case
of zero overlap between initial and final states demonstrating anomalously big
average outputs and sudden jump in time-integrated output. We present and
discuss the numerical evaluation of the probability distribution aiming at
extend- ing the analytic results and describing a realistic experimental
situation of a qubit in the regime of resonant fluorescence.
