On the repeated measurement of continuous observables in quantum mechanics

Abstract We generalize Umegaki's definition of an expectation on the algebra of all bounded operators on a Hilbert space, and classify certain classes of expectations subject to a covariance condition with respect to a unitary representation of a given locally compact group. While Umegaki's expectations only exist for discrete observables, we show that interesting classes of our expectations exist in the general case of continuous observables, and discuss the implications of our work in the theory of quantum measurements.