An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics

We report on an "anti-Gleason" phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a non-linear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commutative functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies in the intersection of two seemingly remote mathematical theories - symplectic topology and the theory of topological quasi-states. We use this quasi-state to estimate the error of the simultaneous measurement of non-commuting Hamiltonians.