An operational approach to quantum probability

In order to provide a mathmatical framework for the process of making repeated measurements on continuous observables in a statistical system we make a mathematical definition of an instrument, a concept which generalises that of an observable and that of an operation. It is then possible to develop such notions as joint and conditional probabilities without any of the commutation conditions needed in the approach via observables. One of the crucial notions is that of repeatability which we show is implicitly assumed in most of the axiomatic treatments of quantum mechanics, but whose abandonment leads to a much more flexible approach to measurement theory.At present on leave from the University of Oxford, Research supported by N.S.F. grant GP-7952X and A.F.O.S.R. contract no. F 44620-67-C-0029.At present on leave from Brasenose College, Oxford. Research supported by A.F.O.S.R. grant AF-AFOFR-69-1712.