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Description

We revisit certain classical and recent results on convergence of averages of a fixed element f of a topological vector space V endowed with an action (g,f)â ¦ áµ f of an amenable (semi)group G. (In the special case when G = â is the semigroup of naturals, the averages are just (¹f + ²f + â ¯ + â ¿f)/n). Such results, collectively called ergodic convergence theoremsâ although there is really nothing â ergodicâ about themâ , include the classical ergodic theorem of Birkhoff as well as von Neumannâ s mean ergodic theorem (MET), alongside subsequent generalizations. In collaboration with J. Iovino, we use continuous logic to obtain a radically elementary proof of a MET valid for any polynomial action of an amenable group on a Hilbert space. The Compactness Theorem from logic implies the existence of universal rates of metastable convergence that depend only on the degree of the action.