## 13 More on hyperarithmetic theory

### 13.1 Hyperarithmetic measure theory

The application of measure theory in recursion theory was initiated by Spector [155], Sacks [117] and Leeuw, Moore, Shannon, and Shapiro [19]. 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 [120]). The set

is .

Proof. By the Spector–Gandy Theorem (3.7.1) it is sufficient to prove that the set

is Σ1 over ...