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We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales. At the same time, we introduce the concept of i.o. subuniformity and relate this concept to recursive measure theory. We prove that there are classes closed downwards under Turing reducibility that have recursive measure zero and that are not i.o. subuniform. This shows that there are examples of classes that cannot be covered with methods other than probabilistic ones. It is easily seen that every set of hyperimmune degree can cover the recursive sets. We prove that there are both examples of hyperimmune-free degree that can and that cannot compute such a cover. [ABSTRACT FROM AUTHOR]

Hjorth and Nies proposed notions of randomness corresponding to the higher recursion setting. In particular, they defined the notion of Π 1 1 -Martin-Löf randomness. In this article we present examples of Π 1 1 -Martin-Löf random reals which are obtained as measures of Π 1 1 open sets. [ABSTRACT FROM AUTHOR]

We investigate the reverse mathematics strength of Martin's pointed tree theorem (MPT) and one of its variants, weak Martin's pointed tree theorem (wMPT). [ABSTRACT FROM AUTHOR]

We elaborate a recursion-theoretic method to define real numbers which are random to any jump of the halting problem. This is obtained by generalizing a result of V. Becher and G. Chaitin dealing with the class of cofinite sets. As in the work of V. Becher and S. Grigorieff (but this time in the context of open sets in Cantor space) appeal to completeness phenomena replaces machine arguments. [ABSTRACT FROM AUTHOR]