 As we have seen, in BDMI and SDMI, the biorthogonal and spectral decompositions do generally not pick out the right properties for the apparatus in non-ideal measurements. However, Bacciagaluppi and Hemmo (1996) show that, when the apparatus is a finite-dimensional system in interaction with an environment containing a large number of degrees of freedom, decoherence guarantees that the spectral decomposition of the reduced apparatus’ state will be very close to the ideally expected result and, as a consequence, an observable very close to and indistinguishable from the apparatus pointer will be selected as definite-valued. Alternatively, Bub (1997) suggests that instead of decoherence—with the “tracing out” of the environment and the diagonalization of the reduced state of the apparatus—the triorthogonal or \(n\)-orthogonal decomposition theorem involving the environment may be used to single out a unique pointer basis for the apparatus.