  Generalized measurements of an observable performed on a quantum system
during a force protocol are investigated and conditions that guarantee the
validity of the Jarzynski equality and the Crooks relation are formulated. In
agreement with previous studies by Campisi {\it et al.} [M. Campisi, P.
Talkner, and P. H\"anggi, Phys. Rev. Lett. {\bf 105}, 140601 (2010); Phys. Rev.
E {\bf 83}, 041114 (2011)], we find that these fluctuation relations are
satisfied for projective measurements; however, for generalized measurements
special conditions on the operators determining the measurements need to be
met. For the Jarzynski equality to hold, the measurement operators of the
forward protocol must be normalized in a particular way. The Crooks relation
additionally entails that the backward and forward measurement operators depend
on each other. Yet, quite some freedom is left as to how the two sets of
operators are interrelated. This ambiguity is removed if one considers
selective measurements, which are specified by a {\it joint} probability
density function of work and measurement results of the considered observable.
We find that the respective forward and backward joint probabilities satisfy
the Crooks relation only if the measurement operators of the forward and
backward protocols are the time-reversed adjoints of each other. In this case,
the work probability density function {\it conditioned} on the measurement
result satisfies a modified Crooks relation. The modification appears as a
protocol-dependent factor that can be expressed by the information gained by
the measurements during the forward and backward protocols. Finally, detailed
fluctuation theorems with an arbitrary number of intervening measurements are
obtained.
