Measure, Integration & Real Analysis, 2021a

258 Chapter 9 Real and Complex Measures
The next definition provides an important class of examples of real and complex
measures.
9.4 measure determined by an L 1 -function
Suppose μ is a (positive) measure on a measurable space (X, S) and h ∈L 1 (μ).
Define ν : S→F by
∫
ν(E) = h dμ.
Then ν is a real measure on (X, S) if F = R and is a complex measure on (X, S)
if F = C.
Proof
( ⋃ ∞
9.5 ν
Suppose E 1 , E 2 ,...is a disjoint sequence of sets in S. Then
k=1
E k
)
=
∫ ( ∞ )
∞ ∫
∑ χ
Ek
(x)h(x) dμ(x) = ∑
k=1
k=1
E
χ
Ek
h dμ =
∞
∑ ν(E k ),
k=1
where the first equality holds because the sets E 1 , E 2 ,...are disjoint and the second
equality follows from the inequality
m
∣ χ
Ek
(x)h(x) ∣ ≤|h(x)|,
∑
k=1
which along with the assumption that h ∈L 1 (μ) allows us to interchange the integral
and limit of the partial sums by the Dominated Convergence Theorem (3.31).
The countable additivity shown in 9.5 means ν is a real or complex measure.
The next definition simply gives a notation for the measure defined in the previous
result. In the notation that we are about to define, the symbol d has no separate
meaning—it functions to separate h and μ.
9.6 Definition h dμ
Suppose μ is a (positive) measure on a measurable space (X, S) and h ∈L 1 (μ).
Then h dμ is the real or complex measure on (X, S) defined by
∫
(h dμ)(E) =
E
h dμ.
Note that if a function h ∈L 1 (μ) takes values in [0, ∞), then h dμ is a finite
(positive) measure.
The next result shows some basic properties of complex measures. No proofs
are given because the proofs are the same as the proofs of the corresponding results
for (positive) measures. Specifically, see the proofs of 2.57, 2.61, 2.59, and 2.60.
Because complex measures cannot take on the value ∞, we do not need to worry
about hypotheses of finite measure that are required of the (positive) measure versions
of all but part (c).
Measure, Integration & Real Analysis, by Sheldon Axler