In mathematics, particularly in functional analysis, a projection-valued measure (or spectral measure) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space.[1] A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.
Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements.[clarification needed] 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.