Spectral measures are an essential tool of functional analysis on Hilbert spaces. Spectral measures are projection-valued measures and are used to state various forms of spectral theorems.
In the following, let $\mathbb{H}$ be a Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathbb{H}$ and $\mathcal{P}(\mathcal{H})$ the orthogonal projections.
The following paragraphs explain the concept of a spectral measure in the real case, sufficient for spectral theorems of selfadjoint operators.
Do not confuse this concept with the partition of unity in differential geometry.
definition: A resolution of the identity operator is a map $E: \mathbb{R} \to \mathcal{P}(\mathcal{H})$ satisfying the following conditions:
(monotony): For $\lambda_1, \lambda_2 \in \mathbb{R}$ with $\lambda_1 \leq \lambda_2$ we have $E(\lambda_1) \leq E(\lambda_2)$.
(continuity from above): for all $\lambda \in \mathbb{R}$ we have $s-\lim_{\epsilon \to 0, \epsilon \gt 0} E(\lambda + \epsilon) = E(\lambda)$.
(boundary condition): $s-\lim_{\epsilon \to -\infty} E(\lambda) = 0$ and $s-\lim_{\epsilon \to \infty} E(\lambda) = \mathbb{1}$.
If there is a finite $\mu \in \mathbb{R}$ such that $E_{\lambda} = 0$ for all $\lambda \leq \mu$ and $E_{\lambda} = \mathbb{1}$ for all $\lambda \geq \mu$, than the resolution is called bounded, otherwise unbounded.
Let E be a spectral resolution and $I$ be a bounded interval in $\mathbb{R}$. The spectral measure of $I$ with respect to $E$ is given by
This allows us to define the integral of a step function $u = \sum_{k=1}^{n} \alpha_k \chi_{I_k}$ with respect to E as
The value of this integral is a bounded operator.
As in conventional measure and integration theory, the integral can be extended from step functions to Borel-measurable functions. In this case one often used notation is
For general function $u, E(u)$ need not be a bounded operator of course, the domain of $E(u)$ is (theorem):
The SNAG theorem is necessary to explain the spectrum condition of the Haag-Kastler axioms.
Let $\mathcal{G}$ be a locally compact, abelian topological group, $\hat \mathcal{G}$ the character group of $\mathcal{G}$, $\mathcal{H}$ a Hilbert space and $\mathcal{U}$ an unitary representation of $\mathcal{G}$ in the algebra of bounded operators of $\mathcal{H}$. The following theorem is sometimes called (classical) SNAG theorem (SNAG = Stone-Naimark-Ambrose-Godement):
The equality holds in the weak sense, i.e. the integral converges in the weak operator topology. The spectrum of $\mathcal{U}(\mathcal{G})$, denoted by $spec\mathcal{U}(\mathcal{G})$, is defined to be the support of this spectral measure $\mathcal{P}$.
The groups of translations $\mathcal{T}$ on $\R^n$ is both isomorph to $\R^n$ and to it’s own character group, every character is of the form $a \mapsto exp(i \langle a, k\rangle)$ for a fixed $k \in \R^n$. So in this case theorem 1 becomes:
This allows us to talk about the support of the spectral measure, i.e. the spectrum of $\mathcal{U}(\mathcal{T})$, as a subset of $\R^n$.
See also projection measure. The theorem 1 is theorem 4.44 in the following classic book: