  Measurements in classical and quantum physics are described in fundamentally
different ways. Nevertheless, one can formally define similar measurement
procedures with respect to the disturbance they cause. Obviously, strong
measurements, both classical and quantum, are invasive -- they disturb the
measured system. We show that it is possible to define general weak
measurements, which are noninvasive: the disturbance becomes negligible as the
measurement strength goes to zero. Classical intuition suggests that
noninvasive measurements should be time symmetric (if the system dynamics is
reversible) and we confirm that correlations are time-reversal symmetric in the
classical case. However, quantum weak measurements -- defined analogously to
their classical counterparts -- can be noninvasive but not time symmetric. We
present a simple example of measurements on a two-level system which violates
time symmetry and propose an experiment with quantum dots to measure the
time-symmetry violation in a third-order current correlation function.
