Random reals as measures of natural open sets

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.