  Performing imperfect or noisy measurements on a quantum system both impacts
the measurement outcome and the state of the system after the measurement. In
this paper we are concerned with imperfect calorimetric measurements. In
calorimetric measurements one typically measures the energy of a thermal
environment to extract information about the system. The measurement is
imperfect in the sense that we simultaneously measure the energy of the
calorimeter and an additional noise bath. Under weak coupling assumptions, we
find that the presence of the noise bath manifests itself by modifying the jump
rates of the reduced system dynamics. We study an example of a driven qubit
interacting with resonant bosons calorimeter and find increasing the noise
leads to a reduction in the power flowing from qubit to calorimeter and thus an
apparent heating up of the calorimeter.
