  We generalize the concept of a weak value of a quantum observable to cover
arbitrary real positive operator measures. We show that the definition is
operationally meaningful in the sense that it can be understood within the
quantum theory of sequential measurements. We then present a detailed analysis
of the recent experiment of Lundeen et al. concerning the reconstruction of the
state of a photon using weak measurements. We compare their method with the
reconstruction method through informationally complete phase space measurements
and show that it lacks the generality of the phase space method. In particular,
a completely unknown state can never be reconstructed using the method of weak
measurements.
