  From the perspective of quantum thermodynamics, realisable measurements cost
work and result in measurement devices that are not perfectly correlated with
the measured systems. We investigate the consequences for the estimation of
work in non-equilibrium processes and for the fundamental structure of the work
fluctuations when one assumes that the measurements are non-ideal. We show that
obtaining work estimates and their statistical moments at finite work cost
implies an imperfection of the estimates themselves: more accurate estimates
incur higher costs. Our results provide a qualitative relation between the cost
of obtaining information about work and the trustworthiness of this
information. Moreover, we show that Jarzynski's equality can be maintained
exactly at the expense of a correction that depends only on the system's energy
scale, while the more general fluctuation relation due to Crooks no longer
holds when the cost of the work estimation procedure is finite. We show that
precise links between dissipation and irreversibility can be extended to the
non-ideal situation.
