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Van Lambalgen's theorem states that a pair $(\alpha,\beta)$ of bitsequencesis Martin-L\"of random if and only if $\alpha$ is Martin-L\"of random and$\beta$ is Martin-L\"of random relative to $\alpha$. In [Information andComputation 209.2 (2011): 183-197, Theorem 3.3], Hayato Takahashi generalizedvan Lambalgen's theorem for computable measures $P$ on a product of two Cantorspaces; he showed that the equivalence holds for each $\beta$ for which theconditional probability $P(\cdot | \beta)$ is computable. He asked whether thiscomputability condition is necessary. We give a positive answer by providing acomputable measure for which van Lambalgen's theorem fails. We also present asimple construction of a measure for which conditional measure is notcomputable. Such measures were first constructed by N. Ackerman, C. Freer andD. Roy in [Proceedings of the 26th Annual IEEE Symposium on Logic in ComputerScience (LICS), pp. 107-116. IEEE (2011)].

Van Lambalgen's theorem fails for some computable measure