  We study the effect of decoherence on a weak value measurement in a paradigm
system consisting of a double quantum dot continuously measured by a quantum
point contact. Fluctuations of the parameters controlling the dot state induce
decoherence. We find that, for measurements longer than the decoherence time,
weak values are always reduced within the range of the eigenvalues of the
measured observable. For measurements at shorter time scales, the measured weak
value strongly depends on the interplay between the decoherence dynamics of the
system and the detector backaction. In particular, depending on the
postselected state and the strength of the decoherence, a more frequent
classical readout of the detector might lead to an enhancement of weak values.
