 In either case, interaction with the environment solves the problem of non-ideal measurements in BDMI and SDMI for the case of finite-dimensional macroscopic apparatuses. However, the case of infinitely many distinct states for the apparatus is more troublesome. Bacciagaluppi (2000) has analyzed this situation, using a continuous model of the apparatus’ interaction with the environment. He concludes that in this case the spectral decomposition of the reduced state of the apparatus generally does not pick out properties that are close enough to empirically required ones. This result seems to apply generally to cases where a macroscopic system with infinitely many degrees of freedom experiences decoherence due to interaction with its environment (see Donald 1998). However, model calculations by Hollowood (2013, 2014) indicate that the problem may be less severe in realistic circumstances than originally supposed, and the issue needs further investigation.