Measure, Integration & Real Analysis, 2021a

270 Chapter 9 Real and Complex Measures
Lebesgue Decomposition Theorem
The next definition captures the notion of one measure having more sets of measure 0
than another measure.
9.32 Definition absolutely continuous; ≪
Suppose ν is a complex measure on a measurable space (X, S) and μ is a
(positive) measure on (X, S). Then ν is called absolutely continuous with respect
to μ, denoted ν ≪ μ,if
ν(E) =0 for every set E ∈Swith μ(E) =0.
9.33 Example absolute continuity
The reader should verify all the following examples:
• If μ is a (positive) measure and h ∈L 1 (μ), then h dμ ≪ μ.
• If ν is a real measure, then ν + ≪|ν| and ν − ≪|ν|.
• If ν is a complex measure, then ν ≪|ν|.
• If ν is a complex measure, then Re ν ≪|ν| and Im ν ≪|ν|.
• Every measure on a measurable space (X, S) is absolutely continuous with
respect to counting measure on (X, S).
The next result should help you think that absolute continuity and singularity are
two extreme possibilities for the relationship between two complex measures.
9.34 absolutely continuous and singular implies 0 measure
Suppose μ is a (positive) measure on a measurable space (X, S). Then the only
complex measure on (X, S) that is both absolutely continuous and singular with
respect to μ is the 0 measure.
Proof Suppose ν is a complex measure on (X, S) such that ν ≪ μ and ν ⊥ μ. Thus
there exist sets A, B ∈Ssuch that A ∪ B = X, A ∩ B = ∅, and ν(E) =ν(E ∩ A)
and μ(E) =μ(E ∩ B) for every E ∈S.
Suppose E ∈S. Then
μ(E ∩ A) =μ ( (E ∩ A) ∩ B ) = μ(∅) =0.
Because ν ≪ μ, this implies that ν(E ∩ A) =0. Thus ν(E) =0. Hence ν is the 0
measure.
Our next result states that a (positive) measure on a measurable space (X, S)
determines a decomposition of each complex measure on (X, S) as the sum of the
two extreme types of complex measures (absolute continuity and singularity).
Measure, Integration & Real Analysis, by Sheldon Axler