{"paper":{"title":"Van Lambalgen's theorem fails for some computable measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Bruno Bauwens","submitted_at":"2015-09-09T18:51:58Z","abstract_excerpt":"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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02884","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}