Projection valued measure

Assume Countable Choice. Let $(X,\mathrm{\Sigma })$ be a measurable space (Measurable spaces and measurable sets, Sigma-algebras) and let $H$ be a complex Hilbert space (Hilbert space). A projection valued measure (PVM) on $(X,\mathrm{\Sigma })$ is a map

$E:\mathrm{\Sigma }⟶\mathcal{B}(H),B⟼E(B),$

such that:

1. $E(B)$ is an orthogonal projection for every $B\in \mathrm{\Sigma }$, that is, a bounded operator with $E(B{)}^2=E(B)=E(B{)}^{\ast }$; there is no finite-dimensional restriction;
2. $E(\mathrm{\varnothing })=0$ and $E(X)=I$;
3. $E(B\cap C)=E(B)E(C)$ for all $B,C\in \mathrm{\Sigma }$;
4. for every pairwise disjoint sequence $(B_n{)}_{n\in \mathbb{N}}$ in $\mathrm{\Sigma }$ with union $B$ and every $x\in H$, the series ${\sum }_{n}E(B_n)x$ converges in norm to $E(B)x$, that is $E(B)x={\sum }_{n=0}^{\mathrm{\infty }}E(B_n)x\mathrm{.}$

Clause 4 is strong countable additivity. A PVM is called regular when $X$ is a locally compact Hausdorff space (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space), the $\sigma$-algebra is the Borel $\sigma$-algebra, and every one of the finite positive measures

$E_x(B):=⟨E(B)x,x⟩,x\in H,B\in \mathrm{\Sigma },$

is a regular Borel measure (Regular Borel measure on an LCH space). Inner products are linear in the first variable and conjugate-linear in the second (Real and complex inner-product spaces and their induced length); this fixed convention is the one used for every pairing $E_{x,y}(B)=⟨E(B)x,y⟩$ on this page.

Well-definedness: the projection clause. The conditions $P^2=P=P^{\ast }$ are simultaneously meaningful and describe exactly the Hilbert orthogonal projections, with no hidden finite-dimensional hypothesis. Indeed, for a bounded $P$ with $P^2=P=P^{\ast }$ the range $\mathrm{ran}P$ is a linear subspace, and for $y=Pw$ one has $⟨x-Px,y⟩=⟨P(x-Px),w⟩=⟨Px-P^{2}x,w⟩=0$, so $x-Px\perp \mathrm{ran}P$; conversely $⟨Pz,Pz⟩=⟨z,P^{\ast }Pz⟩=⟨z,Pz⟩=0$ for $z\perp \mathrm{ran}P$, so $z\in \ker P$, while $⟨z,Px⟩=⟨Pz,x⟩=0$ for $z\in \ker P$ gives $\ker P=(\mathrm{ran}P{)}^{\perp }$ (Hilbert-adjoint identities, Real and complex inner-product spaces and their induced length). Hence $\mathrm{ran}P=(\ker P{)}^{\perp }$ is closed, and the two defining properties $Px\in \mathrm{ran}P$, $x-Px\in (\mathrm{ran}P{)}^{\perp }$ of The Hilbert orthogonal projection onto a closed subspace show that $P=P_{\mathrm{ran}P}$ is the Hilbert orthogonal projection onto a closed subspace, and conversely every such $P_M$ is idempotent and self-adjoint by Hilbert projections are linear, self-adjoint and contractive and Orthogonal decomposition by a closed subspace. Finally $P$ is contractive: from $⟨Px,x⟩=⟨Px,Px⟩=\mathrm{\parallel }Px{\mathrm{\parallel }}^2$ and Cauchy–Schwarz, $\mathrm{\parallel }Px{\mathrm{\parallel }}^2\le \mathrm{\parallel }Px\mathrm{\parallel }\mathrm{\parallel }x\mathrm{\parallel }$, so $\mathrm{\parallel }Px\mathrm{\parallel }\le \mathrm{\parallel }x\mathrm{\parallel }$ for all $x$, and $⟨Px,x⟩=\mathrm{\parallel }Px{\mathrm{\parallel }}^2\ge 0$ is a nonnegative real number.

Well-definedness: the regularity clause. For an orthogonal projection value the pairing $E_x(B)=⟨E(B)x,x⟩$ is a nonnegative real number and $E_x(X)=⟨x,x⟩=\mathrm{\parallel }x{\mathrm{\parallel }}^2<+\mathrm{\infty }$, so every $E_x$ is a finite nonnegative set function and the regularity requirement is a meaningful condition on it; that each $E_x$ is genuinely a countably additive measure of total mass $\mathrm{\parallel }x{\mathrm{\parallel }}^2$, and that the polarized pairings $E_{x,y}$ are finite complex measures, is proved as Scalar and complex measures from a pvm before either is used.