Conditional probabilities and van Lambalgen theorem revisited

Abstract

The definition of conditional probability in case of continuous distributions was an important step in the development of mathematical theory of probabilities. How can we define this notion in algorithmic probability theory? In this survey we discuss the developments in this direction trying to explain what are the difficulties and what can be done to avoid them. Almost all the results discussed in this paper have been published (and we provide the references), but we tried to put them into perspective and to explain the proofs in a more intuitive way. We assume that the reader is familiar with basic notions of measure theory and algorithmic randomness.