Algorithmic randomness over general spaces

Algorithmic randomness over general spaces has been considered such as an effective topological space and a computable metric space. In this paper we generalize algorithmic randomness to a computable topological space. First we define computable measures on a computable topological space and study computability of the evaluation. Next we define randomnesses via three approaches. Measure randomness is defined by tests and has a characterization by martingales. Complexity randomness is define by complexity and is equivalent to measure randomness under two natural conditions. Furthermore van Lambalgen’s theorem holds when these randomnesses coincide.