Lebesgue equivalence

In Section 3.1 we introduce Lebesgue equivalence and define the complex vector space in Section 3.2 we define complex scalar products in... 

With the help of Exercise 3.1 we can see that uniform approximation can be applied to Lebesgue equivalence classes of functions. 

Exercise 3.1 (Used in Section 3.5) In this exercise we show how to make sense of inequalities on Lebesgue equivalence classes of functions. Suppose S is a set with an integral defined on it and f is a real-valued functions on S. Let [] denote the Lebesgue equivalence class of f. We say that [] is strictly positive (0 < []) if for every function fi such that 0 < lA (x) for all x e S, we have... 

Show that the truth of this statement depends only on the equivalence class of f. Show that any inequality (such as [] < e) can be rewritten in the form 0 < something. Thus we can make sense of inequalities of Lebesgue equivalence classes of functions. 

Proof. [Sketch] We leave it to the reader to check the first two criteria of Definition 3.2. As for Criterion 3, positive definiteness follows directly from the definition of the integral, while nondegeneracy can be deduced from the theory of Lebesgue integration, using the first equivalence relation defined in Section 3.1. The interested reader can work out the details in Exercise 3.9 or consult Rudin [Ru74, Theorem 1.39]. ... 

---