Conditional Measure and the Violation of Van Lambalgen’s Theorem for Martin-Löf Randomness

Abstract

Van Lambalgen's theorem states that a pair (ź, β) of bit sequences is Martin-Lof random if and only if ź is Martin-Lof random and β is Martin-Lof random relative to ź. 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(ź|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 computable measure for which conditional measure is not computable. Such measures were first constructed by Ackerman et al. ([1]).