Randomness and initial segment complexity for measures

Abstract

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures.We begin with examples, and provide a robustness property related to Solovay tests. The initial segment complexity of a measure μ at a length n is defined as the μ-average over the descriptive complexity of strings of length n, in the sense of either C or K. We relate this weak randomness notion for a measure to the growth of its initial segment complexity. We show that a maximal growth implies the weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We discuss C-triviality and K-triviality for measures and relate these two notions with each other. Here, triviality means that the initial segment complexity grows as slowly as possible.We show that every measure that is Martin-Löf random in the sense of Hoyrup and Rojas is Martin-Löf absolutely continuous; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures. We conclude with several open questions.