  Van Lambalgen's theorem states that a pair $(\alpha,\beta)$ of bitsequences
is 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 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 $\beta$ for which the
conditional probability $P(\cdot | \beta)$ 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)].
