  For continuous weak measurement of qubits, we obtain exact expressions for
weak values (WVs) from the post-selection restricted average of measurement
outputs, by using both the quantumtrajectory- equation (QTE) and quantum
Bayesian approach. The former is applicable to short-time weak measurement,
while the latter can relax the measurement strength to finite. We find that
even in the "very" weak limit the result can be essentially different from the
one originally proposed by Aharonov, Albert and Vaidman (AAV), in a sense that
our result incorporates non-perturbative correction which could be important
when the AAV's WV is large. Within the Bayesian framework, we obtain also
elegant expressions for finite measurement strength and find that the
amplifier's noise in quantum measurement has no effect on the WVs. In
particular, we obtain very useful result for homodyne measurement in
circuit-QED system, which allows for measuring the real and imaginary parts of
the AAV's WV by simply tuning the phase of the local oscillator. This advantage
can be exploited as efficient state-tomography technique.
