 In quantum mechanics, PVMs are the mathematical description of projective measurements.They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state.A projection-valued measure on a measurable space 
formula_1, where formula_2 is a σ-algebra of subsets of formula_3, formula_4 is a mapping from formula_2 to the set of self-adjoint projections on a Hilbert space formula_6 (i.e.