  Schwinger's algebra of selective measurements has a natural interpretation in
the formalism of groupoids. Its kinematical foundations, as well as the
structure of the algebra of observables of the theory, was presented in two
previous papers (arXiv:1905.12274 and arXiv:1907.03883). In this paper, a
closer look to the statistical interpretation of the theory is taken and it is
found that an interpretation in terms of Sorkin's quantum measure emerges
naturally. It is proven that a suitable class of states of the algebra of
virtual transitions of the theory allows to define quantum measures by means of
the corresponding decoherence functionals. Quantum measures satisfying a
reproducing property are described and a class of states, called factorizable
states, possessing the Dirac-Feynman `exponential of the action' form are
characterized. Finally, Schwinger's transformation functions are interpreted
similarly as transition amplitudes defined by suitable states. The simple
examples of the qubit and the double slit experiment are described in detail,
illustrating the main aspects of the theory.
