Algorithmic tests and randomness with respect to a class of measures

This paper offers some new results on randomness with respect to classes of measures, along with a didactic exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability $p$ of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli measure $B_p$. A notion of “uniform test” for Bernoulli sequences is introduced which allows a quantitative strengthening of this result. Uniform tests are then generalized to arbitrary measures. Bernoulli measures $B_p$ have the important property that $p$ can be recovered from each random sequence of $B_p$. The paper studies some important consequences of this orthogonality property (as well as most other questions mentioned above) also in the more general setting of constructive metric spaces.