  We study a Jarzysnki type equality for work in systems that are monitored
using non-projective unsharp measurements. The information acquired by the
observer from the outcome $f$ of an energy measurement, and the subsequent
conditioned normalized state $\hat \rho(t,f)$ evolved up to a final time $t$
are used to define work, as the difference between the final expectation value
of the energy and the result $f$ of the measurement. The Jarzynski equality
obtained depends on the coherences that the state develops during the process,
the characteristics of the meter used to measure the energy, and the noise it
induces into the system. We analyze those contributions in some detail to
unveil their role. We show that in very particular cases, but not in general,
the effect of such noise gives a factor multiplying the result that would be
obtained if projective measurements were used instead of non-projective ones.
The unsharp character of the measurements used to monitor the energy of the
system, which defines the resolution of the meter, leads to different scenarios
of interest. In particular, if the distance between neighboring elements in the
energy spectrum is much larger than the resolution of the meter, then a similar
result to the projective measurement case is obtained, up to a multiplicative
factor that depends on the meter. A more subtle situation arises in the
opposite case in which measurements may be non-informative, i.e. they may not
contribute to update the information about the system. In this case, a
correction to the relation obtained in the non-overlapping case appears. We
analyze the conditions in which such a correction becomes negligible. We also
study the coherences, in terms of the relative entropy of coherence developed
by the evolved post-measurement state. We illustrate the results by analyzing a
two-level system monitored by a simple meter.
