On construction of positive operator-valued measures possessing projector-valued densities

In non-commutative probability theory a great role is played by positive operator-valued measures (POVM) allowing to restore states (positive nuclear operators) of a system. Among them one can highlight measures possessing projector-valued densities with respect to some scalar measures (discrete or Lebesgue). If projectors are one-dimensional we obtain the task of comparing the properties of the set of vectors and the set of the corresponding one-dimensional projectors. Thus, it appeared that if POVM has a density consisting of one-dimensional projectors, then generating vectors should be linear dependent in some sense. We plan to discuss the problems appearing under a construction.