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. 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. 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.
Definition Let H {\displaystyle H} denote a separable complex Hilbert space and ( X , M ) {\displaystyle (X,M)} a measurable space consisting of a set X {\displaystyle X} and a Borel σ-algebra M {\displaystyle M} on X {\displaystyle X} . A projection-valued measure π {\displaystyle \pi } is a map from M {\displaystyle M} to the set of bounded self-adjoint operators on H {\displaystyle H} satisfying the following properties:
π ( E ) {\displaystyle \pi (E)} is an orthogonal projection for all E ∈ M . {\displaystyle E\in M.}
π ( ∅ ) = 0 {\displaystyle \pi (\emptyset )=0} and π ( X ) = I {\displaystyle \pi (X)=I} , where ∅ {\displaystyle \emptyset } is the empty set and I {\displaystyle I} the identity operator. If E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\dotsc } in M {\displaystyle M} are disjoint, then for all v ∈ H {\displaystyle v\in H} ,
π ( ⋃ j = 1 ∞ E j ) v = ∑ j = 1 ∞ π ( E j ) v . {\displaystyle \pi \left(\bigcup _{j=1}^{\infty }E_{j}\right)v=\sum _{j=1}^{\infty }\pi (E_{j})v.}
π ( E 1 ∩ E 2 ) = π ( E 1 ) π ( E 2 ) {\displaystyle \pi (E_{1}\cap E_{2})=\pi (E_{1})\pi (E_{2})} for all E 1 , E 2 ∈ M . {\displaystyle E_{1},E_{2}\in M.}
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