On-Line Probability, Complexity and Randomness

Classical probability theory considers probability distributions that assign probabilities to all events (at least in the finite case). However, there are natural situations where only part of the process is controlled by some probability distribution while for the other part we know only the set of possibilities without any probabilities assigned. We adapt the notions of algorithmic information theory (complexity, algorithmic randomness, martingales, a priori probability) to this frame work and show that many classical results are still valid