Uniform test of algorithmic randomness over a general space

TLDR: In this article, a general framework for the analysis of randomness in non-compact spaces is introduced, where the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution.

Abstract: The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary distributions (by Levin). We recall the main ideas and problems of Levin's theory, and report further progress in the same framework. - We allow non-compact spaces (like the space of continuous functions, underlying the Brownian motion). - The uniform test (deficiency of randomness) d_P(x) (depending both on the outcome x and the measure P should be defined in a general and natural way. - We see which of the old results survive: existence of universal tests, conservation of randomness, expression of tests in terms of description complexity, existence of a universal measure, expression of mutual information as "deficiency of independence. - The negative of the new randomness test is shown to be a generalization of complexity in continuous spaces; we show that the addition theorem survives. The paper's main contribution is introducing an appropriate framework for studying these questions and related ones (like statistics for a general family of distributions).