Randomness and Initial Segment Complexity for Probability Measures.

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure mu on the space of infinite bit sequences is Martin-Lof absolutely continuous if the non-Martin-Lof random bit sequences form a null set with respect to mu. We think of this as a weak randomness notion for measures. We begin with examples, and a robustness property related to Solovay tests. Our main work connects our property to the growth of the initial segment complexity for measures mu; the latter is defined as a it-average over the complexity of strings of the same length. We show that a maximal growth implies our weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We briefly discuss K-triviality for measures, which means that the growth of initial segment complexity is as slow as possible. We show that full Martin-Lof randomness of a measure implies Martin-Lof absolute continuity; 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.