  The non-selective and selective measurements of a self-adjoint observable $A$
in quantum mechanics are interpreted as `jumps' of the state of the measured
system into a decohered or pure state, respectively, characterized by the
spectral projections of $A$. However, one may try to describe the measurement
results as asymptotic states of a dynamical process, where the non-unitarity of
time evolution arises as an effective description of the interaction of the
measured system with the measuring device. The dynamics we present is a
two-step dynamics: the first step is the non-selective measurement or
decoherence, which is known to be described by the linear, deterministic
Lindblad equation. The second step is a process from the resulted decohered
state to a pure state, which is described by an effective non-linear `randomly
chosen' toy model dynamics: the pure states arise as asymptotic fixed points,
and their emergent probabilities are the relative volumes of their attractor
regions.
