  We study the correspondence between classical and quantum measurements on a
harmonic oscillator that describes a one-mode bosonic field. We connect the
quantum measurement of an observable of the field with the possibility of
amplifying the observable ideally through a quantum amplifier. The
``classical'' measurement corresponds to the joint measurement of the position
$q$ and momentum $p$ of the harmonic oscillator, with following evaluation of a
function $f$ of the outcome $\alpha=q+ip$. For the electromagnetic field the
joint measurement is achieved by a heterodyne detector. The quantum measurement
of an observable $\hat O$ is obtained by preamplifying the heterodyne detector
through an ideal amplifier of $\hat O$, and rescaling the outcome by the gain
$g$. We give a general criterion which states when this preamplified heterodyne
detection scheme approaches the ideal quantum measurement of $\hat O$ in the
limit of infinite gain. We show that this criterion is satisfied and the ideal
measurement is achieved for the case of the photon number operator and for the
quadrature. For both operators the method is robust to nonunit quantum
efficiency of the heterodyne detector. On the other hand, we show that the
preamplified heterodyne detection scheme does not work for arbitrary observable
of the field. As a counterexample, we prove that the simple quadratic function
of the field $\hat K=i(a^{\dag 2}-a^2)/2$ has no corresponding polynomial
function $f(\alpha,\bar \alpha)$---including the obvious choice
$f=\hbox{Im}(\alpha^2)$---that allows the measurement of $\hat K$ through the
preamplified heterodyne measurement scheme.
