  The quantum theory of indirect measurements in physical systems is studied.
The example of an indirect measurement of an observable represented by a
self-adjoint operator $\mathcal{N}$ with finite spectrum is analysed in detail.
The Hamiltonian generating the time evolution of the system in the absence of
direct measurements is assumed to be given by the sum of a term commuting with
$\mathcal{N}$ and a small perturbation not commuting with $\mathcal{N}$. The
system is subject to repeated direct (projective) measurements using a single
instrument whose action on the state of the system commutes with $\mathcal{N}$.
If the Hamiltonian commutes with the observable $\mathcal{N}$ (i.e., if the
perturbation vanishes) the state of the system approaches an eigenstate of
$\mathcal{N}$, as the number of direct measurements tends to $\infty$. If the
perturbation term in the Hamiltonian does \textit{not} commute with
$\mathcal{N}$ the system exhibits "jumps" between different eigenstates of
$\mathcal{N}$. We determine the rate of these jumps to leading order in the
strength of the perturbation and show that if time is re-scaled appropriately a
maximum likelihood estimate of $\mathcal{N}$ approaches a Markovian jump
process on the spectrum of $\mathcal{N}$, as the strength of the perturbation
tends to $0$.
