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 $P(A)=P(A{)}^{*}$

• $P(A_1\cap A_2)=P(A_1)P(A_2)$ for all $A_1,A_2\in B$ where the product is the composition of the operators

• $P(A_1\cup A_2)=P(A_1)+P(A_2)$ 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 $P(A_n)\to P(A)$ in the strong operator topology. (note: check if strong)

Typical example is that $(X,\tau )$ is a topological space and $B$ is the $\sigma$-algebra $ℬ(X)$ 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