  With the purpose of controlling the steady state of a dielectric nanosphere
levitated within an optical cavity, we study its conditional dynamics under
simultaneous sideband cooling and additional time-continuous measurement of
either the output cavity mode or the nanosphere's position. We find that the
average phonon number, purity and quantum squeezing of the steady-states can
all be made more non-classical through the addition of time-continuous
measurement. We predict that the continuous monitoring of the system, together
with Markovian feedback, allows one to stabilize the dynamics for any value of
the laser frequency driving the cavity. By considering state-of-the-art values
of the experimental parameters, we prove that one can in principle obtain a
non-classical (squeezed) steady-state with an average phonon number $n_{\sf
ph}\approx 0.5$.
