  The structure theorem is established which shows that an arbitrary multi-mode
bosonic Gaussian observable can be represented as a combination of four basic
cases, the physical prototypes of which are homodyne and heterodyne, noiseless
or noisy, measurements in quantum optics. The proof establishes connection
between the description of Gaussian observable in terms of the characteristic
function and in terms of density of the probability operator-valued measure
(POVM) and has remarkable parallels with treatment of bosonic Gaussian channels
in terms of their Choi-Jamiolkowski form. Along the way we give the ``most
economical'', in the sense of minimal dimensions of the quantum ancilla,
construction of the Naimark extension of a general Gaussian observable. It is
also shown that the Gaussian POVM has bounded operator-valued density with
respect to the Lebesgue measure if and only if its noise covariance matrix is
nondegenerate.
