Measure, Integration & Real Analysis, 2021a

Section 9B Decomposition Theorems 279
10 Suppose μ is a (positive) measure on a measurable space (X, S) and ν is a
complex measure on (X, S). Show that the following are equivalent:
(a) ν ≪ μ.
(b) |ν| ≪μ.
(c) Re ν ≪ μ and Im ν ≪ μ.
11 Suppose μ is a (positive) measure on a measurable space (X, S) and ν is a real
measure on (X, S). Show that ν ≪ μ if and only if ν + ≪ μ and ν − ≪ μ.
12 Suppose μ is a (positive) measure on a measurable space (X, S). Prove that
is a closed subspace of M F (S).
{ν ∈M F (S) : ν ≪ μ}
13 Give an example to show that the Radon–Nikodym Theorem (9.36) can fail if
the σ-finite hypothesis is eliminated.
14 Suppose μ is a (positive) σ-finite measure on a measurable space (X, S) and ν
is a complex measure on (X, S). Show that the following are equivalent:
(a) ν ≪ μ.
(b) for every ε > 0, there exists δ > 0 such that |ν(E)| < ε for every set
E ∈Swith μ(E) < δ.
(c) for every ε > 0, there exists δ > 0 such that |ν|(E) < ε for every set
E ∈Swith μ(E) < δ.
15 Prove 9.42 [with the extra hypothesis that μ is a σ-finite (positive) measure] in
the case where p = 1.
16 Explain where the proof of 9.42 fails if p = ∞.
17 Prove that if μ is a (positive) measure and 1 < p < ∞, then L p (μ) is reflexive.
[See the definition before Exercise 19 in Section 7B for the meaning of reflexive.]
18 Prove that L 1 (R) is not reflexive.
Measure, Integration & Real Analysis, by Sheldon Axler