  In the present paper we consider the problem of description of an arbitrary
generalized quantum measurement with outcomes in a measurable space. Analyzing
the unitary invariants of a measuring process, we present the most general form
of a possible integral representation of an instrument, which differs from the
representations of an instrument available in the mathematical and physical
literature. We introduce the notion of a quantum stochastic representation of
an instrument, whose elements are wholly determined by the unitary invariants
of a measuring process. We show that the description of a generalized direct
quantum measurement can be considered in the frame of a new general approach,
which we call the quantum stochastic approach (QSA), based on the notion of a
family of quantum stochastic evolution operators, satisfying the orthonormality
relation and describing the conditional evolution of a quantum system under a
measurement. The QSA allows to give: a) the complete statistical description of
any generalized direct quantum measurement (a POV measure and a family of
posterior states); b) the complete description in a Hilbert space of the
stochastic behaviour of a quantum system under a measurement in the sense of
specification of the probabilistic transition law governing the change from the
initial state of a quantum system to a final one under a single measurement; c)
to formalize the consideration of all possible types of quantum measurements.
For measurements continuous in time the QSA allows, in particular, to define in
the most general case (without assuming any Markov property) the notion of
posterior pure state trajectories (quantum trajectories) and to give their
probabilistic treatment.
