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The study in hand deals with a conjecture of Sarnak from 2010 about the orthogonality of sequences induced by deterministic dynamical systems to the Möbius function. Its main results are the ergodic theorem with Möbius weights, which is a measure theoretic (weaker) version of Sarnak's conjecture, and the already assured validity of Sarnak's conjecture in special cases, where we have exemplarily chosen the Thue-Morse shift and skew product extensions of rational rotations on the circle et al.