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In this paper we investigate algorithmic randomness on more general spacesthan the Cantor space, namely computable metric spaces. To do this, we firstdevelop a unified framework allowing computations with probability measures. Weshow that any computable metric space with a computable probability measure isisomorphic to the Cantor space in a computable and measure-theoretic sense. Weshow that any computable metric space admits a universal uniform randomnesstest (without further assumption).

Computability of probability measures and Martin-Lof randomness over metric spaces