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Keywords Analytic sets Automatic continuity Effective convergence rates. Effectively open set Ergodic Theorem Layerwise Computable Lusin's Theorem Martin-Lof random Polish group, Non-singular transformation co-analytic sets collapsing hierarchy computability computable analysis effective infinitesimals expansions of the real ordered additive group, fractals, monadic second order theory of one successor infinite measure-preserving transformation intuitionistic separation theorems nonstandard analysis rates of convergence recursive inequalities subgradient methods