646 CHAPTER 23. REPRESENTATION THEOREMS

12. Suppose λ (E) =∫

E hdµ where h is real valued and µ is a finite measure so that λ

is also real valued. Let P,N be a Hahn decomposition for λ . Show that |λ |(E) =∫E |h|dµ. Hint: Argue that on P it follows h ≥ 0 a.e. and on N,h ≤ 0 a.e. Then

estimate ∑F∈π(E) λ (F) using a Hahn decomposition. If we defined |x+ iy|1 as |x|1+|y|1 , and the total variation exactly the same way for a complex valued measure ex-cept for letting |·|1 refer to this way of measuring magnitude, then everything wouldbe much easier. Why don’t we do this and save a lot of trouble?