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.

Let denote a separable complex Hilbert space and a measurable space consisting of a set and a Borel σ-algebra on . A **projection-valued measure** is a map from to the set of bounded self-adjoint operators on satisfying the following properties:^{[2]}^{[3]}

- is an orthogonal projection for all
- and , where is the empty set and the identity operator.
- If in are disjoint, then for all ,

- for all

The second and fourth property show that if and are disjoint, i.e., , the images and are orthogonal to each other.

Let and its orthogonal complement denote the image and kernel, respectively, of . If is a closed subspace of then can be wrtitten as the *orthogonal decomposition* and is the unique identity operator on satisfying all four properties.^{[4]}^{[5]}

For every and the projection-valued measure forms a complex-valued measure on defined as

with total variation at most .^{[6]} It reduces to a real-valued measure when

and a probability measure when is a unit vector.

**Example** Let be a *σ*-finite measure space and, for all , let

be defined as

i.e., as multiplication by the indicator function on *L*^{2}(*X*). Then defines a projection-valued measure.^{[6]} For example, if , , and there is then the associated complex measure which takes a measurable function and gives the integral

If π is a projection-valued measure on a measurable space (*X*, *M*), then the map

extends to a linear map on the vector space of step functions on *X*. In fact, it is easy to check that this map is a ring homomorphism. This map extends in a canonical way to all bounded complex-valued measurable functions on *X*, and we have the following.

**Theorem** — For any bounded Borel function on , there exists a unique bounded operator such that
^{[7]}^{[8]}

where is a finite Borel measure given by

Hence, is a finite measure space.

The theorem is also correct for unbounded measurable functions but then will be an unbounded linear operator on the Hilbert space .

This allows to define the Borel functional calculus for such operators and then pass to measurable functions via the Riesz–Markov–Kakutani representation theorem. That is, if is a measurable function, then a unique measure exists such that

See also: Self-adjoint operator § Spectral theorem |

Let be a separable complex Hilbert space, be a bounded self-adjoint operator and the spectrum of . Then the spectral theorem says that there exists a unique projection-valued measure , defined on a Borel subset , such that^{[9]}

where the integral extends to an unbounded function when the spectrum of is unbounded.^{[10]}

First we provide a general example of projection-valued measure based on direct integrals. Suppose (*X*, *M*, μ) is a measure space and let {*H*_{x}}_{x ∈ X } be a μ-measurable family of separable Hilbert spaces. For every *E* ∈ *M*, let π(*E*) be the operator of multiplication by 1_{E} on the Hilbert space

Then π is a projection-valued measure on (*X*, *M*).

Suppose π, ρ are projection-valued measures on (*X*, *M*) with values in the projections of *H*, *K*. π, 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 π 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 {*H*_{x}}_{x ∈ X }, such that π is unitarily equivalent to multiplication by 1_{E} on the Hilbert space

The measure class^{[clarification needed]} of μ and the measure equivalence class of the multiplicity function *x* → dim *H*_{x} completely characterize the projection-valued measure up to unitary equivalence.

A projection-valued measure π is *homogeneous of multiplicity* *n* if and only if the multiplicity function has constant value *n*. Clearly,

**Theorem**. Any projection-valued measure π taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:

where

and

See also: Expectation value (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 π expresses the probability that the observable takes on various values.

A common choice for *X* is the real line, but it may also be

**R**^{3}(for position or momentum in three dimensions ),- a discrete set (for angular momentum, energy of a bound state, etc.),
- the 2-point set "true" and "false" for the truth-value of an arbitrary proposition about Φ.

Let *E* be a measurable subset of the measurable space *X* and Φ a normalized vector-state in *H*, so that its Hilbert norm is unitary, ||Φ|| = 1. The probability that the observable takes its value in the subset *E,* given the system in state Φ, is

where the latter notation is preferred in physics.

We can parse this in two ways.

First, for each fixed *E*, the projection π(*E*) is a self-adjoint operator on *H* whose 1-eigenspace is the states Φ for which the value of the observable always lies in *E*, and whose 0-eigenspace is the states Φ for which the value of the observable never lies in *E*.

Second, for each fixed normalized vector state , the association

is a probability measure on *X* making the values of the observable into a random variable.

A measurement that can be performed by a projection-valued measure π is called a **projective measurement**.

If *X* is the real number line, there exists, associated to π, a Hermitian operator *A* defined on *H* by

which takes the more readable form

if the support of π is a discrete subset of **R**.

The above operator A is called the observable associated with the spectral measure.

The idea of a projection-valued measure is generalized by the positive operator-valued measure (POVM), where the need for the orthogonality implied by projection operators is replaced by the idea of a set of operators that are a non-orthogonal partition of unity^{[clarification needed]}. This generalization is motivated by applications to quantum information theory.