The Random Members of a Π10 Class

We examine several notions of randomness for elements in a given Π 1 0 class P . Such an effectively closed subset P of 2 ω may be viewed as the set of infinite paths through the tree T P of extendible nodes of P , i.e., those finite strings that extend to a member of P , so one approach to defining a random member of P is to randomly produce a path through T P using a sufficiently random oracle for advice. In addition, this notion of randomness for elements of P may be induced by a map from 2 ω onto P that is computable relative to T P , and the notion even has a characterization in term of Kolmogorov complexity. Another approach is to define a relative measure on P by conditionalizing the Lebesgue measure on P , which becomes interesting if P has Lebesgue measure 0. Lastly, one can alternatively define a notion of incompressibility for members of P in terms of the amount of branching at levels of T P . We explore some notions of homogeneity for Π 1 0 classes, inspired by work of van Lambalgen. A key finding is that in a specific class of sufficiently homogeneous Π 1 0 classes P , all of these approaches coincide. We conclude with a discussion of random members of Π 1 0 classes of positive measure.