Measure, Integration & Real Analysis, 2021a

262 Chapter 9 Real and Complex Measures
To prove the inequality above in the other direction, suppose E 1 ,...,E n ∈Sare
disjoint sets such that E 1 ∪···∪E n ⊂ ⋃ ∞
k=1
A k . Then
∞
∑
k=1
|ν|(A k ) ≥
=
≥
=
∞ n
∑ ∑
k=1 j=1
n ∞
∑ ∑
j=1 k=1
n
∑
j=1
∣
n
∑
j=1
∞
∑
k=1
|ν(E j ∩ A k )|
|ν(E j ∩ A k )|
|ν(E j )|,
ν(E j ∩ A k ) ∣
where the first line above follows from the definition of |ν|(A k ) and the last line
above follows from the countable additivity of ν.
The inequality above and the definition of |ν| ( ⋃ ∞k=1
A k
)
imply that
completing the proof.
∞
∑
k=1
( ⋃ ∞
|ν|(A k ) ≥|ν|
k=1
A k
),
The Banach Space of Measures
In this subsection, we make the set of complex or real measures on a measurable
space into a vector space and then into a Banach space.
9.13 Definition addition and scalar multiplication of measures
Suppose (X, S) is a measurable space. For complex measures ν, μ on (X, S)
and α ∈ F, define complex measures ν + μ and αν on (X, S) by
(ν + μ)(E) =ν(E)+μ(E) and (αν)(E) =α ( ν(E) ) .
You should verify that if ν, μ, and α are as above, then ν + μ and αν are complex
measures on (X, S). You should also verify that these natural definitions of addition
and scalar multiplication make the set of complex (or real) measures on a measurable
space (X, S) into a vector space. We now introduce notation for this vector space.
9.14 Definition M F (S)
Suppose (X, S) is a measurable space. Then M F (S) denotes the vector space
of real measures on (X, S) if F = R and denotes the vector space of complex
measures on (X, S) if F = C.
Measure, Integration & Real Analysis, by Sheldon Axler