Editing Projection-valued measure (section)

{{Short description|Mathematical operator-value measure of interest in quantum mechanics and functional analysis}}
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]] [[projection (mathematics)|projection]]s on a fixed [[Hilbert space]].{{sfn | Conway | 2000 | p=41}} A projection-valued measure (PVM) is formally similar to a [[real-valued]] [[Measure (mathematics)|measure]], except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to [[integration (mathematics)|integrate]] [[complex-valued function]]s 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#Spectral subspaces and projection-valued measures|spectral theorem]] for [[Self-adjoint operator#Formulation in the physics literature|self-adjoint operators]], in which case the PVM is sometimes referred to as the [[Spectral theory of ordinary differential equations#Spectral measure|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 [[Quantum measurement|projective measurements]].{{clarify|reason=Is this a novel term? It's not defined in the linked article.|date=May 2015}}  They are generalized by [[POVM|positive operator valued measures]] (POVMs) in the same sense that a [[mixed state (physics)|mixed state]] or [[density matrix]] generalizes the notion of a [[pure state]].