Algorithmic randomness over general spaces

The study of Martin-Lof randomness on a computable metric space with a computable measure has seen much progress recently. In this paper we study Martin-Lof randomness on a more general space, that is, a computable topological space with a computable measure. On such a space, Martin-Lof randomness may not be a natural notion because there is no universal test, and Martin-Lof randomness and complexity randomness (defined in this paper) do not coincide in general. We show that SCT3 is a sufficient condition for the existence and coincidence, and study how much we can weaken this condition.

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