  The most general type of measurement in quantum physics is modeled by a
positive operator-valued measure (POVM). Mathematically, a POVM is a
generalization of a measure, whose values are not real numbers, but positive
operators on a Hilbert space. POVMs can equivalently be viewed as maps between
effect algebras or as maps between algebras for the Giry monad. We will show
that this equivalence is an instance of a duality between two categories. In
the special case of continuous POVMs, we obtain two equivalent representations
in terms of morphisms between von Neumann algebras.
