The Random Members of a Π 1 0 ${\Pi }_{1}^{0}$ Class

We examine several notions of randomness for elements in a given Π10 class P. Such an effectively closed subset P of 2ωmay be viewed as the set of infinite paths through the tree TP 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 TP using a sufficiently random oracle for advice. In addition, this notion of randomness for elements of P may be induced by a map from 2ωonto P that is computable relative to TP, 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 TP. We explore some notions of homogeneity for Π10 classes, inspired by work of van Lambalgen. A key finding is that in a specific class of sufficiently homogeneous Π10 classes P, all of these approaches coincide. We conclude with a discussion of random members of Π10 classes of positive measure.