Measure, Integration & Real Analysis, 2021a

266 Chapter 9 Real and Complex Measures
5 Suppose μ is a (positive) measure on a measurable space (X, S) and h is a
nonnegative function in L 1 (μ). Let ν be the (positive) measure on (X, S)
defined by dν = h dμ. Prove that
∫ ∫
f dν = fhdμ
for all S-measurable functions f : X → [0, ∞].
6 Suppose (X, S, μ) is a (positive) measure space. Prove that
is a closed subspace of M F (S).
{h dμ : h ∈L 1 (μ)}
7 (a) Suppose B is the collection of Borel subsets of R. Show that the Banach
space M F (B) is not separable.
(b) Give an example of a measurable space (X, S) such that the Banach space
M F (S) is infinite-dimensional and separable.
8 Suppose t > 0 and λ is Lebesgue measure on the σ-algebra of Borel subsets of
[0, t]. Suppose h : [0, t] → C is the function defined by
h(x) =cos x + i sin x.
Let ν be the complex measure defined by dν = h dλ.
(a) Show that ‖ν‖ = t.
(b) Show that if E 1 , E 2 ,...is a sequence of disjoint Borel subsets of [0, t], then
∞
∑
k=1
|ν(E k )| < t.
[This exercise shows that the supremum in the definition of |ν|([0, t]) is not
attained, even if countably many disjoint sets are allowed.]
9 Give an example to show that 9.9 can fail if the hypothesis that ν is a real
measure is replaced by the hypothesis that ν is a complex measure.
10 Suppose (X, S) is a measurable space with S ̸= {∅, X}. Prove that the total
variation norm on M F (S) does not come from an inner product. In other
words, show that there does not exist an inner product 〈·, ·〉 on M F (S) such
that ‖ν‖ = 〈ν, ν〉 1/2 for all ν ∈M F (S), where ‖·‖ is the usual total variation
norm on M F (S).
11 For (X, S) a measurable space and b ∈ X, define a finite (positive) measure δ b
on (X, S) by
{
1 if b ∈ E,
δ b (E) =
0 if b /∈ E
for E ∈S.
(a) Show that if b, c ∈ X, then ‖δ b + δ c ‖ = 2.
(b) Give an example of a measurable space (X, S) and b, c ∈ X with b ̸= c
such that ‖δ b − δ c ‖ ̸= 2.
Measure, Integration & Real Analysis, by Sheldon Axler