Editing Projection-valued measure (section)

== Definition ==
Let <math>H</math> denote a [[separable space|separable]] [[complex number|complex]] [[Hilbert space]] and  <math>(X, M)</math> a [[measurable space]] consisting of a set <math>X</math> and a [[Borel_set|Borel σ-algebra]]  <math>M</math> on <math>X</math>. A '''projection-valued measure''' <math>\pi</math> is a map from <math>M</math> to the set of [[Self-adjoint_operator#Bounded_self-adjoint_operators|bounded self-adjoint operators]] on <math>H</math> satisfying the following properties:{{sfn | Hall | 2013 | p=138}}{{sfn | Reed | Simon | 1980 | p=234}}
* <math>\pi(E)</math> is an [[Projection_(linear_algebra)#Orthogonal_projections|orthogonal projection]] for all <math>E \in M.</math>
* <math>\pi(\emptyset) = 0</math> and <math>\pi(X) = I</math>, where <math>\emptyset</math> is the [[empty set]] and <math>I</math> the [[identity operator]].
* If <math>E_1, E_2, E_3,\dotsc</math> in <math>M</math> are disjoint, then for all <math>v \in H</math>,
::<math>\pi\left(\bigcup_{j=1}^{\infty} E_j \right)v = \sum_{j=1}^{\infty} \pi(E_j) v.</math>
* <math>\pi(E_1 \cap E_2)= \pi(E_1)\pi(E_2)</math> for all <math>E_1, E_2 \in M.</math>

The second and fourth property show that if <math>
E_1 </math> and <math>E_2</math> are disjoint, i.e., <math>E_1 \cap E_2 = \emptyset</math>, the images <math>\pi(E_1)</math> and <math>\pi(E_2)</math> are [[orthogonal]] to each other. 

Let <math>V_E = \operatorname{im}(\pi(E))</math> and its [[orthogonal complement]] <math>V^\perp_E=\ker(\pi(E))</math> denote the [[Image_(mathematics)|image]] and  [[Kernel_(linear_algebra)|kernel]], respectively, of <math>\pi(E)</math>. If <math>V_E </math> is a closed subspace of <math>H</math> then <math>H</math> can be wrtitten as the  ''orthogonal decomposition'' <math>H=V_E \oplus V^\perp_E</math> and <math>\pi(E)=I_E</math> is the unique identity operator on <math>V_E </math> satisfying all four properties.{{sfn | Rudin | 1991 | p=308}}{{sfn | Hall | 2013 | p=541}}

For every <math>\xi,\eta\in H</math> and <math>E\in M</math> the projection-valued measure forms a [[complex measure|complex-valued measure]] on <math>H</math> defined as
:<math> \mu_{\xi,\eta}(E) := \langle \pi(E)\xi \mid \eta \rangle 
</math>
with [[total variation]] at most <math>\|\xi\|\|\eta\|</math>.{{sfn | Conway | 2000 | p=42}} It reduces to a real-valued [[Measure_(mathematics)|measure]] when
:<math> \mu_{\xi}(E) := \langle \pi(E)\xi \mid \xi \rangle 
</math>
and a [[probability measure]] when <math>\xi</math> is a [[unit vector]].

'''Example'''  Let <math>(X, M, \mu)</math> be a [[Measure space#Important classes of measure spaces|{{math|''&sigma;''}}-finite measure space]] and, for all <math>E \in M</math>, let 
:<math>
\pi(E) : L^2(X) \to L^2 (X)
</math>
be defined as
:<math>\psi \mapsto \pi(E)\psi=1_E \psi,</math>
i.e., as multiplication by the [[indicator function]] <math>1_E</math> on [[Lp space|''L''<sup>2</sup>(''X'')]]. Then <math>\pi(E)=1_E</math> defines a projection-valued measure.{{sfn | Conway | 2000 | p=42}} For example, if <math>X = \mathbb{R}</math>, <math>E = (0,1)</math>, and <math>\varphi,\psi \in L^2(\mathbb{R})</math> there is then the associated complex measure <math>\mu_{\varphi,\psi}</math> which takes a measurable function <math>f: \mathbb{R} \to \mathbb{R}</math> and gives the integral
:<math>\int_E f\,d\mu_{\varphi,\psi} = \int_0^1 f(x)\psi(x)\overline{\varphi}(x)\,dx</math>