A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM).

The concept of "measurement operator," which was introduced in that work was an early version of what is currently referred to as positive-operator valued measure (POVM) in the context of quantum measurement theory.

In functional analysis and quantum measurement theory, a positive-operator valued measure (POVM) is a measure whose values are non-negative self-adjoint operators on a Hilbert space, and whose integral is the identity operator.