Van Lambalgen's Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen's Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of the reals in the range of Omega operators. It is known that is high. We extend this result to that is . We also prove that there exists A such that, for each n, the real is for some universal Turing machine M by using the extended van Lambalgen's Theorem.
Kenshi Miyabe. "An Extension of van Lambalgen's Theorem to Infinitely Many Relative 1-Random Reals." Notre Dame J. Formal Logic 51 (3) 337 - 349, 2010. https://doi.org/10.1215/00294527-2010-020