  Nanomechanical oscillators have been employed as transducers to measure
force, mass and charge with high sensitivity. They are also used in opto- or
electromechanical experiments with the goal of quantum control and phenomena of
mechanical systems. Here, we report the realization and operation of a hybrid
monolithically integrated transducer system consisting of a high-$Q$
nanomechanical oscillator with modes in the MHz regime coupled to the
near-field of a high-$Q$ optical whispering-gallery-mode microresonator. The
transducer system enables a sensitive resolution of the nanomechanical beam's
thermal motion with a signal-to-noise of five orders of magnitude and has a
force sensitivity of $74\,\rm{aN}\,\rm{Hz}^{-1/2}$ at room temperature. We
show, both theoretically and experimentally, that the sensitivity of continuous
incoherent force detection improves only with the fourth root of the averaging
time. Using dissipative feedback based on radiation pressure enabled control,
we explicitly demonstrate by detecting a weak incoherent force that this
constraint can be significantly relaxed. We achieve a more than 30-fold
reduction in averaging time with our hybrid transducer and are able to detect
an incoherent force having a force spectral density as small as
$15\,\rm{aN}\,\rm{Hz}^{-1/2}$ within $35\,\rm{s}$ of averaging. This
corresponds to a signal which is 25 times smaller than the thermal noise and
would otherwise remain out of reach. The reported monolithic platform is an
enabling step towards hybrid nanomechanical transducers relying on the
light-mechanics interface.
