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We report on an "anti-Gleason" phenomenon in classical mechanics: in contrastwith the quantum case, the algebra of classical observables can carry anon-linear quasi-state, a monotone functional which is linear on all subspacesgenerated by Poisson-commutative functions. We present an example of such aquasi-state in the case when the phase space is the 2-sphere. This example liesin the intersection of two seemingly remote mathematical theories - symplectictopology and the theory of topological quasi-states. We use this quasi-state toestimate the error of the simultaneous measurement of non-commutingHamiltonians.Comment: Revised version: discussions on Gleason's theorem and the classical  limit of a quantum spin system added, bibliography updated. To appear in  Foundations of Physic

An “Anti-Gleason” Phenomenon and Simultaneous Measurements in Classical Mechanics