Using Problem 6 complete the Hahn decomposition for λ having values in (−∞,∞].
Now the Hahn Jordan decomposition for the measure λ is
Explain why λ− is a finite measure. Hint: Let N0 = ∅. For Nn a given negative set,
Explain why you can assume that for all n, tn < 0. Let En ⊆ NnC such that
and from Problem 6 let An ⊆ En be a negative set such that λ
≤ λ. Then
Nn+1 ≡ Nn ∪ An. If tn does not converge to 0 explain why there exists a set having
measure −∞ which is not allowed. Thus tn → 0. Let N = ∪n=1∞Nn and explain why
P ≡ NC must be positive due to tn → 0.