A weak randomness notion for probability measures.

We study probability measures on Cantor space, thinking of them as statistical superpositions of bit sequences. We say that a measure on the space of infinite bit sequences is Martin-Loef absolutely continuous if the non-Martin-Loef random bit sequences form a null set under this measure. We analyse this notion as a weak randomness notion for measures. We begin with examples and a robustness property related to Solovay test. Then we study the growth of initial segment complexity for measures (defined as a average under the measure over the complexity of strings of the same length) and relate it to our weak randomness property. We introduce K-triviality for measures. We seek an appropriate effective version of the Shannon-McMillan-Breiman theorem where the trajectories are replaced by measures.