In quantum mechanics, POVMs can be considered more fundamental than PVMs because PVMs can be considered a special sort of POVM. However, because of the Naimark extension, one can formally consider POVMs to always be derived from PVMs. Accordingly, one could argue that choosing PVMs or POVMs to be fundamental is a matter of taste. From a Copenhagenish standpoint, POVMs are more fundamental, since there is no wave function of the universe and the extension to a larger Hilbert space is not always physical. However, one interesting explanation for the emergence of a preferred basis is decoherence. Formally, this depends on a unique basis for the Schmidt decomposition when systems+apparatus+environment is considered (reviewed by Schlosshauer in http://arxiv.org/abs/quant-ph/0312059). The Schmidt decomposition is an orthogonal decomposition, so it corresponds to a basis for a PVM. Does this mean that PVMs are more fundamental than POVMs from the standpoint of decoherence?