  It has recently been established that, in a non-demolition measurement of an
observable $\mathcal{N}$ with a finite point spectrum, the density matrix of
the system approaches an eigenstate of $\mathcal{N}$, i.e., it "purifies" over
the spectrum of $\mathcal{N}$. We extend this result to observables with
general spectra. It is shown that the spectral density of the state of the
system converges to a delta function exponentially fast, in an appropriate
sense. Furthermore, for observables with absolutely continuous spectra, we show
that the spectral density approaches a Gaussian distribution over the spectrum
of $\mathcal{N}$. Our methods highlight the connection between the theory of
non-demolition measurements and classical estimation theory.
