A remarkable achievement in algorithmic randomness and algorithmic information theory was the discovery of the notions of Ktrivial, K-low and Martin-Löf-random-low sets: three different definitions turn out to be equivalent for very non-trivial reasons [1,3,5]. This survey, based on the course taught by one of the authors (L.B.) in Poncelet laboratory (CNRS, Moscow) in 2014, provides an exposition of the proof of this equivalence and some related results. We assume that the reader is familiar with basic notions of algorithmic information theory (see, e.g.,  for introduction and  for more detailed exposition). More information about the subject and its history can be found in [2,6].
Bienvenu, L., & Shen, A. (2015). K-trivial, K-low and MLR-low sequences: A tutorial. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9300, pp. 1–23). Springer Verlag. https://doi.org/10.1007/978-3-319-23534-9_1