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Preprint Number 1126

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1126. Eduardo Dueñez and José N. Iovino
Model theory and metric convergence I: Metastability and dominated convergence

Submission date: 30 November 2016


We study Tao's finitary viewpoint of convergence in metric spaces, as captured by the notion of metastability with a uniform rate. We adopt the perspective of continuous model theory. We show that, in essence, uniform metastability is the only formulation of metric convergence that can be captured by a theory in continuous first-order logic, a result we call the Uniform Metastability Principle. Philosophically, this principle amounts to the following meta-theorem: “If a classical statement about convergence in metric structures is refined to a statement about uniformly metastable convergence, then the validity of the original statement implies the validity of its uniformly metastable version.” As an instance of this phenomenon, we formulate an abstract version of Tao's Metastable Dominated Convergence Theorem as a statement about axiomatizable classes of metric structures, and show that it is a direct consequence of the Uniform Metastability Principle.

Mathematics Subject Classification: 40A05 (Primary) 03Cxx, 46Bxx, 28-xx (Secondary)

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Full text arXiv 1611.09971: pdf, ps.

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