  We revisit aspects of monitoring observables with continuous spectrum in a
quantum system subject to dissipative (Lindbladian) or conservative
(Hamiltonian) evolutions. After recalling some of the salient features of the
case of pure monitoring, we deal with the case when monitoring is in
competition with a Lindbladian evolution. We show that the strong measurement
limit leads to a diffusion on the spectrum of the observable. For the case with
competition between observation and Hamiltonian dynamics, we exhibit a scaling
limit in which the crossover between the classical regime and a diffusive
regime can be analyzed in details.
