We investigate different geometries and invariant measures on the space of
mixed Gaussian quantum states. We show that when the global purity of the state
is held fixed, these measures coincide and it is possible, within this constraint,
to define a unique notion of volume on the space of mixed Gaussian states.
We then use the so defined measure to study typical non-classical correlations
of two mode mixed Gaussian quantum states, in particular entanglement and
steerability. We show that under the purity constraint alone, typical values
for symplectic invariants can be computed very elegantly, irrespectively of
the non-compactness of the underlying state space. Then we consider finite
volumes by constraining the purity and energy of the Gaussian state and
compute typical values of quantum correlations numerically.