Andrei Kolmogorov and Leonid Levin on Randomness

This chapter provides a translation of a letter written in 1939, in which Andrei Kolmogorov’s explains his views on the connection between probability theory and its applications. It also provides some historical sources on the early history of Kolmogorov complexity in Russia (then the USSR): abstracts of talks by Kolmogorov and letters from Leonid Levin to Kolmogorov. The relation between Levin’s measures and semimeasures on the one hand and martingales and supermartingales on the other is explained in the present volume’s chapter “Martingales in the Study of Randomness”.