---
tag: RDE658H8Z4
kind: definition
name: Projection-valued measure
msc: [47B15, 28B05, 46C05]
defines: ATGT25E5FF
uses: [HM2HHE3J51, 0B8KZRT28M, NY6TYHRGMS, 0SV8K0TDGY]
sources:
  - url: https://en.wikipedia.org/wiki/Projection-valued_measure
---

## Definition

Let $(\Omega, \Sigma)$ be a measurable space, so that $\Sigma$ is a [[sigma-algebra|0SV8K0TDGY]] of subsets of $\Omega$.
Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] with identity map $I$.
A *projection-valued measure* on $(\Omega, \Sigma)$ is a map $E$ from $\Sigma$ to the bounded operators on $H$ such that:

1. $E(B)$ is an [[orthogonal projection|0B8KZRT28M]] for every $B \in \Sigma$;
2. $E(\emptyset) = 0$ and $E(\Omega) = I$;
3. $E(B \cap C) = E(B) E(C)$ for all $B, C \in \Sigma$;
4. for all $x, y \in H$, the map $E_{x,y}(B) = \ip{E(B) x}{y}$ is a [[complex measure|NY6TYHRGMS]] on $\Sigma$.

## Remarks

Clause 4 is equivalent to the strong countable additivity: if $B = \bigcup_n B_n$ with $B_n \in \Sigma$ pairwise disjoint, then $E(B) x = \sum_n E(B_n) x$ for every $x \in H$.
By clause 3, the projections of disjoint sets have orthogonal images, and all $E(B)$ commute.
For each $x$, $E_{x,x}$ is a positive measure of total mass $\norm{x}^2$.
Example: let $H = \ell^2$ with orthonormal basis $(e_n)_{n \ge 1}$, let $\Omega = \R$ with its Borel sets, and let $(\lambda_n)$ be real numbers.
Let $E(B)$ be the orthogonal projection onto the closed span of $\set{e_n : \lambda_n \in B}$; clauses 1 to 4 follow from the fact that $E_{x,y}(B) = \sum_{n : \lambda_n \in B} x_n \overline{y_n}$.
Non-example: on $\Omega = \R$, let $E(B) = P$ if $0 \in B$ and $E(B) = 0$ otherwise, for an orthogonal projection $P \ne I$; then $E(\R) = P \ne I$, so clause 2 fails.