Rank and Randomness.

We show that for each computable ordinal it is possible to find in each Martin-Lof random degree a sequence R of Cantor-Bendixson rank alpha, while ensuring that the sequences that inductively witness R's rank are all Martin-Lof 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.