bounded linear operator such that for all where denotes the complex measure from the definition of . The map is a homomorphism of rings. An integral notation is often used for , as in The theorem is also correct for unbounded measurable functions f, but then will be an unbounded linear operator on the Hilbert space H. The spectral theorem says that every self-adjoint operator has an associated projection-valued measure defined on the real axis, such that This allows to define the Borel functional calculus for such operators: if is a measurable function, we set Structure of projection-valued measures First we provide a general example of projection-valued measure based on direct integrals. Suppose (X, M, μ) is a measure space and let {Hx}x ∈ X be a μ-measurable family of separable Hilbert spaces. For every E ∈ M, let (E) be the operator of multiplication by 1E on the Hilbert space Then is a projection-valued measure on (X, M). Suppose , ρ are projection-valued measures on (X, M) with values in the projections of H, K. , ρ are unitarily equivalent if and only if there is a unitary operator U:H → K such that for every E ∈ M. Theorem. If (X, M) is a standard Borel space, then for every projection-valued measure on (X, M) taking values in the projections of a separable Hilbert space, there is a Borel measure μ and a μ-measurable family of Hilbert spaces {Hx}x ∈ X , such that is unitarily equivalent to multiplication by 1E on the Hilbert space The measure class of μ and the measure equivalence class of the multiplicity function x → dim Hx completely characterize the projection-valued measure up to unitary equivalence. A projection-valued measure is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly, Theorem. Any projection-valued measure taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures: where and Application in quantum mechanics In quantum mechanics, given a projection valued measure of a measurable space X to the space of continuous endomorphisms upon a Hilbert space H, the projective space of the Hilbert space H is interpreted as the set of possible states Φ of a quantum system, the measurable space X is the value space for some quantum property of the system (an "observable"), the projection-valued measure expresses the