A generalization of the functional calculus of observables and notion of joint measurability to the case of non-commuting observables

For any pair of bounded observables A and B with pure point spectra, we construct an associated 'joint observable' which gives rise to a notion of a joint (projective) measurement of A and B, and which conforms to the intuition that one can measure non-commuting observables simultaneously, provided one is willing to give up arbitrary precision. As an application, we show how our notion of a joint observable naturally allows for a construction of a 'functional calculus,' so that for any pair of observables A and B as above, and any (Borel measurable) function f : R-2 -> R, a new 'generalized observable' f (A, B) is obtained. Moreover, we show that this new functional calculus has some rather remarkable properties.