Effective Randomness for Computable Probability Measures

Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of random elements coincide. Elaborating on work of Bienvenu [Bienvenu, L., Constructive equivalence relations on computable probability measures, International Computer Science Symposium in Russia, Lecture Notes in Computer Science 3967 (2006), 92–103], we determine all the implications that hold between the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness, and weak randomness, and the equivalence and consistency relations of probability measures, except that we do not know whether two computable probability measures need to be equivalent in case their respective concepts of weakly randomness coincide.