[Wikipedia](https://en.wikipedia.org/wiki/POVM) gives the following definition of *positive operator-valued measure* (POVM):

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> 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 $M$.
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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?