# nLab projection measure

Projection (or: projection-valued) measures are operator-valued measures of a special type. They appear for example in the theory of reproducing kernel Hilbert spaces, coherent states and the foundations of quantum mechanics. A projection measure is used to parametrize a complete family of projection operators by subsets of some parameter space.

Given a set $X$ and some $\sigma$-algebra $B$ of subsets of $X$, with $X\in B$, and a complex Hilbert space $H$, a map $P:B\to \mathrm{End}H$ is called a projection-(valued) measure on $B$ with values in $\mathrm{End}H$ if

• all operators in the image are selfadjoint $P\left(A\right)=P\left(A{\right)}^{*}$

• $P\left({A}_{1}\cap {A}_{2}\right)=P\left({A}_{1}\right)P\left({A}_{2}\right)$ for all ${A}_{1},{A}_{2}\in B$ where the product is the composition of the operators

• $P\left({A}_{1}\cup {A}_{2}\right)=P\left({A}_{1}\right)+P\left({A}_{2}\right)$ for all ${A}_{1},{A}_{2}\in B$ such that ${A}_{1}\cap {A}_{2}=\varnothing$

• if ${A}_{n}\to A$, in the sense of coinciding upper and lower limit of sets, $A={\cap }_{n}{\cup }_{k\ge n}{A}_{k}={\cup }_{n}{\cap }_{k\ge n}{A}_{k}$, then $P\left({A}_{n}\right)\to P\left(A\right)$ in the strong operator topology. (note: check if strong)

Typical example is that $\left(X,\tau \right)$ is a topological space and $B$ is the $\sigma$-algebra $ℬ\left(X\right)$ of Borel subsets of $X$.

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995, Zbl
• A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988
Revised on November 29, 2013 11:52:39 by Zoran Škoda (161.53.130.104)