General definition of POVM

Question

Wikipedia gives the following definition of positive operator-valued measure (POVM):

A POVM on a measurable space $(X,M)$ is a function $F$ defined on $M$ whose values are bounded non-negative self-adjoint operators on a Hilbert space $\mathcal{H}$ such that $F(X) = 1_\mathcal{H}$ and for every $\psi \in \mathcal{H}$ the map $E \mapsto \langle F(E) \psi , \psi \rangle$ is a non-negative countably additive measure on $M$.

When the set of outcomes $X$ is finite and/or the Hilbert space has finite dimension, a simpler definition can be easily found in standard textbooks, but in the general case none of the books I've checked (on quantum mechanics, functional analysis or $C^*$-algebras) contains a definition. Do you have a reference, other than Wikipedia, for an equivalent definition of POVM?

Answer 1

Here are two books which feature a definition of POVMs as general as what you are looking for that immediately came to mind:

• "Quantum Theory of Open Systems" (1976) by Davies: he introduces POVMs in Chapter 4, Definition 1.1 (although he called it "PMV")
• "The Mathematical Language of Quantum Theory" (2012) by Heinosaari & Ziman: see Definition 3.5 therein

Answer 2

One possible source is "Notes on Spectral Theory" by Berberian. He defines POVMs in Definition 1. Note that strictly speaking, his definition is not equivalent to Wikipedia's definition since he only assumes the measure to be defined on a ring of subsets instead of a $\sigma$-algebra and he does not require the normalization $F(X)=1_{\mathcal H}$. But if the domain happens to be a $\sigma$-algebra and $F(X)=1_{\mathcal H}$, both definitions coincide (see Theorem 1).