Measure, Integration & Real Analysis, 2021a

268 Chapter 9 Real and Complex Measures
(n+1 ⋃
ν
j=k
) ( ⋃ n
A j = ν
≥
j=k
(
a −
= a −
) ( ( ⋃ n
A j + ν(A n+1 ) − ν
n
∑
j=k
n+1
∑
j=k
j=k
1
) (
2 j + a − 1 )
2 n+1 − a
1
2 j ,
A j
) ∩ An+1
)
where the first line follows from 9.7(b) and the second line follows from 9.25 and
9.26. We have now verified that 9.26 holds if n is replaced by n + 1, completing the
proof by induction of 9.26.
The sequence of sets A k , A k ∪ A k+1 , A k ∪ A k+1 ∪ A k+2 ,...is increasing. Thus
taking the limit as n → ∞ of both sides of 9.26 and using 9.7(c) gives
( ⋃ ∞
9.27 ν
Now let
j=k
A j
)
≥ a − 1
2 k−1 .
∞⋂ ∞⋃
A = A j .
k=1 j=k
The sequence of sets ⋃ ∞
j=1 A j , ⋃ ∞
j=2 A j ,...is decreasing. Thus 9.27 and 9.7(d) imply
that ν(A) ≥ a. The definition of a now implies that
ν(A) =a.
Suppose E ∈Sand E ⊂ A. Then ν(A) =a ≥ ν(A \ E). Thus we have
ν(E) =ν(A) − ν(A \ E) ≥ 0, which proves (b).
Let B = X \ A; thus (a) holds. Suppose E ∈Sand E ⊂ B. Then we have
ν(A ∪ E) ≤ a = ν(A). Thus ν(E) =ν(A ∪ E) − ν(A) ≤ 0, which proves (c).
Jordan Decomposition Theorem
You should think of two complex or positive measures on a measurable space (X, S)
as being singular with respect to each other if the two measures live on different sets.
Here is the formal definition.
9.28 Definition singular measures; ν ⊥ μ
Suppose ν and μ are complex or positive measures on a measurable space (X, S).
Then ν and μ are called singular with respect to each other, denoted ν ⊥ μ, if
there exist sets A, B ∈Ssuch that
• A ∪ B = X and A ∩ B = ∅;
• ν(E) =ν(E ∩ A) and μ(E) =μ(E ∩ B) for all E ∈S.
Measure, Integration & Real Analysis, by Sheldon Axler