13 More on hyperarithmetic theory
13.1 Hyperarithmetic measure theory
The application of measure theory in recursion theory was initiated by Spector , Sacks  and Leeuw, Moore, Shannon, and Shapiro . Randomness theory on the natural numbers may be viewed as such an application, while higher randomness theory is the union of measure theory with hyperarithmetic theory. In this chapter we discuss these connections to set the stage for higher randomness in the next chapter.
13.1.1 The measure of -sets
Lemma 13.1.1 (Sacks ). The set
Proof. By the Spector–Gandy Theorem (3.7.1) it is sufficient to prove that the set
is Σ1 over ...