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Keywords Analytic sets Borel-Cantelli lemma Effective convergence rates. Effectively open set Erdős–Rényi Theorem Ergodic Theorem Kochen-Stone theorem Layerwise Computable Locally compact abelian group, Pontryagin–van Kampen duality, Fourier transform, nonstandard analysis, hyperfinite, infinitesimal, approximation Lusin's Theorem Martin-Lof random Polish group, Non-singular transformation co-analytic sets collapsing hierarchy computability computable analysis integration theory, measure theory, non-Archimedean mathematics intuitionistic separation theorems metastability model theory, von Neumann algebra, free entropy, random matrix, strongly 1-bounded, ultraproduct o-minimal structures, log-analytic functions, exponential number, preparation theorems