N. L. Ackerman, C. E. Freer, and D. M. Roy, Noncomputable conditional distributions, LICS, pp.107-116, 2011.

N. L. Ackerman, C. E. Freer, and D. M. Roy, On computability and disintegration, Mathematical Structures in Computer Science, pp.1-28, 2016.

M. Archibald, V. Brattka, and C. Heuberger, Randomness with respect to the signed-digit representation, Fundam. Inform, vol.83, issue.1-2, pp.1-19, 2008.

E. Asarin and A. Pokrovskii, Use of the Kolmogorov complexity in analyzing control system dynamics, Avtomatika i Telemekhanika, vol.47, issue.1, p.1, 1986.

J. Avigad, Uncomputably noisy ergodic limits, Notre Dame Journal of Formal Logic, vol.53, issue.3, pp.347-350, 2012.

J. Avigad, E. T. Dean, and J. Rute, Algorithmic randomness, reverse mathematics, and the dominated convergence theorem, Ann. Pure Appl. Logic, vol.163, issue.12, pp.1854-1864, 2012.

J. Avigad, P. Gerhardy, and H. Towsner, Local stability of ergodic averages, Transactions of the American Mathematical Society, vol.362, pp.261-288, 2010.

L. M. Axon, Algorithmically random closed sets and probability, 2010.

L. M. Axon, Martin-Löf randomness in spaces of closed sets, The Journal of Symbolic Logic, vol.80, issue.2, pp.359-383, 2015.

G. Barmpalias, P. Brodhead, D. Cenzer, S. Dashti, and R. Weber, Algorithmic randomness of closed sets, J. Log. and Comput, vol.17, issue.6, pp.1041-1062, 2007.

B. Bauwens, Conditional measure and the violation of van Lambalgen's theorem for Martin-Löf randomness. Theory of Computing Systems, pp.1-10, 2016.

V. Becher and S. Figueira, An example of a computable absolutely normal number, Theor. Comput. Sci, vol.270, issue.1-2, pp.947-958, 2002.

L. Bienvenu, K. Allen, and T. A. Slaman, On zeros of Martin-Löf random Brownian motion, J. Logic & Analysis, vol.6, 2014.

L. Bienvenu, A. Day, I. Mezhirov, and A. Shen, Ergodic-type characterizations of algorithmic randomness, CiE, Lecture Notes in Computer Science, vol.6158, pp.49-58, 2010.

L. Bienvenu, A. R. Day, M. Hoyrup, I. Mezhirov, and A. Shen, A constructive version of Birkhoff's ergodic theorem for Martin-Löf random points, Information and Computation, vol.210, pp.21-30, 2012.

L. Bienvenu, P. Gács, M. Hoyrup, C. Rojas, and A. Shen, Algorithmic tests and randomness with respect to a class of measures, Proceedings of the Steklov Institute of Mathematics, vol.274, issue.1, pp.34-89, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00644785

L. Bienvenu and W. Merkle, Effective randomness for computable probability measures. Electron, Notes Theor. Comput. Sci, vol.167, pp.117-130, 2007.

L. Bienvenu and B. Monin, Von Neumann's biased coin revisited, Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science, LICS 2012, pp.145-154, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01397207

E. Bishop, Foundations of constructive analysis. McGraw-Hill series in higher mathematics, 1967.

V. Bosserhoff, Are unbounded linear operators computable on the average for Gaussian measures, Journal of Complexity, vol.24, issue.4, pp.477-491, 2008.

V. Bosserhoff, Notions of probabilistic computability on represented spaces, Journal of Universal Computer Science, vol.14, issue.6, pp.956-995, 2008.

V. Brattka, G. Gherardi, and R. Hölzl, Las Vegas computability and algorithmic randomness, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, vol.30, pp.130-142, 2015.

V. Brattka, G. Gherardi, and R. Hlzl, Probabilistic computability and choice, Information and Computation, vol.242, pp.249-286, 2015.

V. Brattka, J. S. Miller, and A. Nies, Randomness and differentiability, Trans. Amer. Math. Soc, vol.368, issue.1, pp.581-605, 2016.

M. Brin and G. Stuck, Introduction to Dynamical Systems, 2002.

P. Brodhead, D. Cenzer, and J. B. Remmel, Random continuous functions, Electron. Notes Theor. Comput. Sci, vol.167, pp.275-287, 2007.

G. Davie, The Borel-Cantelli lemmas, probability laws and Kolmogorov complexity, Annals of Probability, vol.29, issue.4, pp.1426-1434, 2001.

G. Davie and W. L. Fouché, On the computability of a construction of Brownian motion, Mathematical Structures in Computer Science, vol.23, issue.6, pp.1257-1265, 2013.

O. Demuth, The differentiability of constructive functions of weakly bounded variation on pseudo numbers, Comment. Math. Univ. Carolin, vol.16, issue.3, pp.583-599, 1975.

D. Diamondstone and B. Kjos-hanssen, Members of random closed sets, Mathematical Theory and Computational Practice, 5th Conference on Computability in Europe, vol.5635, pp.144-153, 2009.

J. Doob, Measure theory. Graduate texts in mathematics, 1994.

R. Downey and D. Hirschfeldt, Algorithmic Randomness and Complexity, 2010.

A. Edalat, LICS '96: Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science, p.372, 1996.

A. Edalat, A computable approach to measure and integration theory, LICS '07: Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science, pp.463-472, 2007.

H. Férée and M. Ziegler, Mathematical Aspects of Computer and Information Sciences: 6th International Conference, pp.489-504, 2015.

W. L. Fouché, Arithmetical representations of Brownian motion I, J. Symb. Log, vol.65, issue.1, pp.421-442, 2000.

W. L. Fouché, The descriptive complexity of Brownian motion, Advances in Mathematics, vol.155, pp.317-343, 2000.

W. L. Fouché, Dynamics of a generic Brownian motion: Recursive aspects, Theor. Comput. Sci, vol.394, issue.3, pp.175-186, 2008.

W. L. Fouché, Mathematical Theory and Computational Practice, 5th Conference on Computability in Europe, vol.5635, pp.208-217, 2009.

W. L. Fouché, Diophantine properties of Brownian motion: recursive aspects, Logic, Computation, Hierarchies, vol.4, pp.139-156, 2014.

W. L. Fouché, Kolmogorov complexity and the geometry of Brownian motion, Mathematical Structures in Computer Science, vol.25, issue.7, pp.1590-1606, 2015.

W. L. Fouché, S. Mukeru, and G. Davie, Fourier spectra of measures associated with algorithmically random Brownian motion, Logical Methods in Computer Science, vol.10, issue.3, 2014.

J. N. Franklin, N. Greenberg, J. S. Miller, and K. M. Ng, Martin-Löf random points satisfy Birkhoff's ergodic theorem for effectively closed sets, Proc. Amer. Math. Soc, vol.140, issue.10, pp.3623-3628, 2012.

J. N. Franklin and H. Towsner, Randomness and non-ergodic systems, Mosc. Math. J, vol.14, issue.4, pp.711-744, 2014.

C. E. Freer, B. Kjos-hanssen, A. Nies, and F. Stephan, Algorithmic aspects of Lipschitz functions, Computability, vol.3, issue.1, pp.45-61, 2014.

C. E. Freer and D. M. Roy, Computable exchangeable sequences have computable de Finetti measures, Lecture Notes in Computer Science, vol.5635, pp.218-231, 2009.

C. E. Freer and D. M. Roy, Computable de Finetti measures, Annals of Pure and Applied Logic, vol.163, issue.5, pp.530-546, 2009.

H. Friedman, The computational complexity of maximization and integration, Advances in Mathematics, vol.53, issue.1, pp.80-98, 1984.

P. Gács, Uniform test of algorithmic randomness over a general space, Theoretical Computer Science, vol.341, pp.91-137, 2005.

P. Gács, M. Hoyrup, and C. Rojas, Randomness on computable probability spaces -A dynamical point of view, STACS, LIPIcs, pp.469-480, 2009.

P. Gács, M. Hoyrup, and C. Rojas, Randomness on computable probability spaces -a dynamical point of view, Theory of Computing Systems, vol.48, issue.3, pp.465-485, 2011.

S. Galatolo, M. Hoyrup, and C. Rojas, A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties, Theoretical Computer Science, vol.410, pp.2207-2222, 2009.

S. Galatolo, M. Hoyrup, and C. Rojas, Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems, Proceedings Seventh International Conference on Computability and Complexity in Analysis, vol.24, pp.7-18, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00517372

S. Galatolo, M. Hoyrup, and C. Rojas, Effective symbolic dynamics, random points, statistical behavior, complexity and entropy. Information and Computation, vol.208, issue.1, pp.23-41, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00517382

S. Galatolo, M. Hoyrup, and C. Rojas, Dynamics and abstract computability: computing invariant measures, Discrete and Continuous Dynamical Systems -A, vol.29, issue.1, pp.193-212, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00517367

S. Galatolo, M. Hoyrup, and C. Rojas, Statistical properties of dynamical systems -simulation and abstract computation, Chaos, Solitons and Fractals, vol.45, issue.1, pp.1-14, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00644790

P. Hertling and Y. Wang, Invariance properties of random sequences, Journal of Universal Computer Science, vol.3, issue.11, pp.1241-1249, 1997.

P. Hertling and K. Weihrauch, Randomness spaces, 25th International Colloquium, ICALP'98, vol.1443, pp.796-807, 1998.

P. Hertling and K. Weihrauch, Random elements in effective topological spaces with measure, Information and Computation, vol.181, issue.1, pp.32-56, 2003.

M. Hochman, Upcrossing inequalities for stationary sequences and applications, The Annals of Probability, vol.37, issue.6, pp.2135-2149, 2009.

R. Hölzl and P. Shafer, Universality, optimality, and randomness deficiency, Ann. Pure Appl. Logic, vol.166, issue.10, pp.1049-1069, 2015.

M. Hoyrup, Randomness and the ergodic decomposition, Lecture Notes in Computer Science, vol.6735, pp.122-131, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00586736

M. Hoyrup, The dimension of ergodic random sequences, STACS. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2012.
URL : https://hal.archives-ouvertes.fr/inria-00606457

M. Hoyrup, Computability of the ergodic decomposition, Annals of Pure and Applied Logic, vol.164, issue.5, pp.542-549, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00746473

M. Hoyrup, Irreversible computable functions, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), vol.25, pp.362-373, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00915952

M. Hoyrup and C. Rojas, An application of Martin-Löf randomness to effective probability theory, Lecture Notes in Computer Science, vol.5635, pp.260-269, 2009.

M. Hoyrup and C. Rojas, Applications of effective probability theory to Martin-Löf randomness, Lecture Notes in Computer Science, vol.5555, pp.549-561, 2009.

M. Hoyrup and C. Rojas, Computability of probability measures and Martin-Löf randomness over metric spaces, Information and Computation, vol.207, issue.7, pp.830-847, 2009.

M. Hoyrup, C. Rojas, and K. Weihrauch, Computability of the Radon-Nikodym derivative, CiE, Lecture Notes in Computer Science, vol.6735, pp.132-141, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00726044

M. Hoyrup, C. Rojas, and K. Weihrauch, Computability of the Radon-Nikodym derivative, Computability, vol.1, issue.1, pp.3-13, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00726044

T. Jafarikhah and K. Weihrauch, The Riesz representation operator on the dual of C[0; 1] is computable, J. UCS, vol.19, issue.6, pp.750-770, 2013.

S. Kakutani and K. Petersen, The speed of convergence in the ergodic theorem, Monatshefte für Mathematik, vol.91, issue.1, pp.11-18, 1981.

A. Kawamura, Computational complexity in analysis and geometry, 2011.

A. Kawamura and S. A. Cook, Complexity theory for operators in analysis, ACM Transactions on Computation Theory, vol.4, issue.2, p.5, 2012.

B. Kjos-hanssen, Infinite subsets of random sets of integers, Mathematical Research Letters, vol.16, pp.103-110, 2009.

B. Kjos-hanssen, The probability distribution as a computational resource for randomness testing, Journal of Logic & Analysis, vol.2, pp.1-13, 2010.

B. Kjos-hanssen and A. Nerode, The law of the iterated logarithm for algorithmically random Brownian motion, LFCS, pp.310-317, 2007.

B. Kjos-hanssen, P. K. Nguyen, and J. M. Rute, Algorithmic randomness for Doob's martingale convergence theorem in continuous time, Logical Methods in Computer Science, vol.10, issue.4, 2014.

K. I. Ko, Complexity Theory of Real Functions, 1991.

M. Van-lambalgen, Random sequences, 1987.

L. A. Levin, Uniform tests of randomness, Soviet Math. Dokl, vol.17, issue.2, pp.337-340, 1976.

M. Li and P. M. Vitanyi, An Introduction to Kolmogorov Complexity and Its Applications, 1993.

H. Lu and K. Weihrauch, Computable Riesz representation for the dual of

, Math. Log. Q, vol.53, issue.4-5, pp.415-430, 2007.

H. Lu and K. Weihrauch, Computable Riesz representation for locally compact hausdorff spaces, J. UCS, vol.14, issue.6, pp.845-860, 2008.

P. Martin-löf, The definition of random sequences, Information and Control, vol.9, issue.6, pp.602-619, 1966.

K. Miyabe, The difference between optimality and universality, Logic Journal of IGPL, 2011.

K. Miyabe, L 1 -computability, layerwise computability and Solovay reducibility, Computability, vol.2, issue.1, pp.15-29, 2013.

K. Miyabe, Algorithmic randomness over general spaces, Mathematical Logic Quarterly, vol.60, issue.3, pp.184-204, 2014.

K. Miyabe, A. Nies, and J. Zhang, Using almost-everywhere theorems from analysis to study randomness, The Bulletin of Symbolic Logic, vol.22, issue.3, pp.305-331, 2016.

K. Miyabe and J. Rute, Van Lambalgen's theorem for uniformly relative Schnorr and computable randomness, 2017.

P. Moser, On the convergence of Fourier series of computable lebesgue integrable functions, Proceedings of the Fourth International Conference on Computability and Complexity in Analysis, vol.202, pp.13-18, 2007.

N. T. Müller, Computability on random variables, Theor. Comput. Sci, vol.219, issue.1-2, pp.287-299, 1999.

M. Nakamura, Ergodic theorems for algorithmically random sequences, Proceedings of the Symposium on Information Theory and Its Applications, vol.28, pp.71-74, 2005.

S. Nandakumar, An effective ergodic theorem and some applications, STOC '08: Proceedings of the 40th annual ACM symposium on Theory of computing, pp.39-44, 2008.

A. Nies, Computability and randomness. Oxford logic guides, 2009.

A. Nies, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), vol.25, pp.602-613, 2014.

N. Pathak, C. Rojas, and S. G. Simpson, Schnorr randomness and the Lebesgue differentiation theorem, Proc. Amer. Math. Soc, vol.142, issue.1, pp.335-349, 2014.

R. R. Phelps, Lectures on Choquet's theorem, vol.London, 1966.

M. B. Pour-el and I. Richards, L p -computability in recursive analysis, Proceedings of the American Mathematical Society, vol.92, issue.1, pp.93-97, 1984.

M. B. Pour-el and J. I. Richards, Computability in Analysis and Physics. Perspectives in Mathematical Logic, 1989.

A. Y. Rumyantsev and A. Shen, Probabilistic constructions of computable objects and a computable version of Lovász local lemma, Fundam. Inform, vol.132, issue.1, pp.1-14, 2014.

J. Rute, Computable randomness and betting for computable probability spaces, Math. Log. Q, vol.62, issue.4-5, pp.335-366, 2016.

J. Rute, When does randomness come from randomness? Theor, Comput. Sci, vol.635, pp.35-50, 2016.

J. Rute, Schnorr randomness for noncomputable measures, Inf. Comput, vol.258, pp.50-78, 2018.

N. Sanin, Constructive Real Numbers and Constructive Function Spaces, Translations of Mathematical Monographs, vol.21, 1968.

C. P. Schnorr, Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie, Lecture Notes in Mathematics, vol.218, 1971.

M. Schröder, Admissible representations of probability measures. Electronic Notes in Theoretical, Computer Science, vol.167, pp.61-78, 2007.

B. Spitters, A constructive view on ergodic theorems, J. Symbolic Logic, vol.71, issue.2, pp.611-623, 2006.

H. Takahashi, On a definition of random sequences with respect to conditional probability, Information and Computation, vol.206, pp.1375-1382, 2008.

H. Takahashi, Algorithmic randomness and monotone complexity on product space. Information and Computation, vol.209, pp.183-197, 2011.

V. G. Vovk, The law of the iterated logarithm for random Kolmogorov, or chaotic, sequences. Theory of Probability and Applications, vol.32, pp.413-425, 1987.

V. V. V'yugin, Effective convergence in probability and an ergodic theorem for individual random sequences, SIAM Theory of Probability and Its Applications, vol.42, issue.1, pp.39-50, 1997.

V. V. V'yugin, Ergodic theorems for individual random sequences, Theoretical Computer Science, vol.207, issue.4, pp.343-361, 1998.

V. V. V'yugin, Non-robustness property of the individual ergodic theorem, Problems of Information Transmission, vol.37, issue.2, pp.27-39, 2001.

K. Weihrauch, Computability on the probability measures on the Borel sets of the unit interval, Theoretical Computer Science, vol.219, pp.421-437, 1999.

K. Weihrauch, Computable Analysis, 2000.

K. Weihrauch, On computable metric spaces Tietze-Urysohn extension is computable, Computability and Complexity in Analysis: 4th International Workshop, pp.357-368, 2000.

K. Weihrauch, Computability on measurable functions, Computability, vol.6, issue.1, pp.79-104, 2017.

K. Weihrauch and N. R. Tavana, Representations of measurable sets in computable measure theory, Logical Methods in Computer Science, vol.10, issue.3, 2014.

Y. Wu and D. Ding, Computability of measurable sets via effective topologies, Archive for Mathematical Logic, vol.45, pp.365-379, 2006.

Y. Wu and K. Weihrauch, A computable version of the Daniell-Stone theorem on integration and linear functionals, Theor. Comput. Sci, vol.359, issue.1-3, pp.28-42, 2006.

B. Y. Zhang and N. Zhong, L p -computability, Math. Log. Q, vol.45, pp.449-456, 1999.