Van Lambalgen's theorem fails for some computable measure

Abstract

Van Lambalgen's theorem states that a pair $(α,β)$ of bitsequences is Martin-Löf random if and only if $α$ is Martin-Löf random and $β$ is Martin-Löf random relative to $α$. 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 $β$ for which the conditional probability $P(\cdot | β)$ is computable. He asked whether this computability condition is necessary. We give a positive answer by providing a computable measure for which van Lambalgen's theorem fails. We also present a simple construction of a measure for which conditional measure is not computable. Such measures were first constructed by N. Ackerman, C. Freer and D. Roy in [Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 107-116. IEEE (2011)].