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- W2224242236 abstract "Van Lambalgen's theorem states that a pair (; ) of bitsequences is Martin- Lof random if and only if is Martin-Lof random and is Martin-Lof random relative to . The denition of Martin-Lof randomness can be generalized to dene randomness relative to any computable measureP . In (Information and Computation 209.2 (2011): 183-197, Theorem 3.3), Hayato Takahashi generalized van Lambalgen's theorem for computable measures P on a product of two Cantor spaces; he showed that the equivalence holds for each for which the conditional probability P (j ) is computable. In this note we show that this additional condition on can not be removed by providing a measure for which van Lambalgen's theorem fails. Michiel van Lambalgen characterized Martin-Lof randomness of a pair of sequences: Theorem 1 ((4)). The following are equivalent for a pair (; ) of sequences: (; ) is Martin-Lof random, is Martin-Lof random and is Martin-Lof random relative to . One can replace Lebesgue measure in the denition of Martin-Lof randomness, by any other computable measure P. We call sequences that are random in this sense P-random. Similar for a pair of measures. To generalize the theorem above for computable bivariate measuresP , we need to x the conditional measure carefully." @default.
- W2224242236 created "2016-06-24" @default.
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- W2224242236 date "2015-09-09" @default.
- W2224242236 modified "2023-09-27" @default.
- W2224242236 title "Van Lambalgen's theorem fails for some computable measure" @default.
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