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Keywords Analytic sets Automatic continuity 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 Quantitative analysis co-analytic sets collapsing hierarchy computability computable analysis infinite measure-preserving transformation integration theory, measure theory, non-Archimedean mathematics intuitionistic separation theorems metastability