Rank and randomness

We show that for each computable ordinal $\alpha >0$ it is possible to find in each Martin-L"of random ${\mathrm{\Delta }}_2^0$ degree a sequence $R$ of Cantor-Bendixson rank $\alpha$, while ensuring that the sequences that inductively witness $R$’s rank are all Martin-L"of 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.