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Rank and randomness

We show that for each computable ordinal $α>0$ it is possible to find in each Martin-Löf random $Δ^0_2$ degree a sequence $R$ of Cantor-Bendixson rank $α$, while ensuring that the sequences that inductively witness $R$'s rank are all Martin-Löf random with respect to a single countably supported and computable measure. This is a strengthening for random degrees of a recent result of Downey, Wu, and Yang, and can be understood as a randomized version of it.