Measure, Integration & Real Analysis, 2021a

Section 9B Decomposition Theorems 269
9.29 Example singular measures
Suppose λ is Lebesgue measure on the σ-algebra B of Borel subsets of R.
• Define positive measures ν, μ on (R, B) by
ν(E) =|E ∩ (−∞,0)| and μ(E) =|E ∩ (2, 3)|
for E ∈B. Then ν ⊥ μ because ν lives on (−∞,0) and μ lives on [0, ∞).
Neither ν nor μ is singular with respect to λ.
• Let r 1 , r 2 ,...be a list of the rational numbers. Suppose w 1 , w 2 ,...is a bounded
sequence of complex numbers. Define a complex measure ν on (R, B) by
w
ν(E) =
k
2 k
∑
{k∈Z + : r k ∈E}
for E ∈B. Then ν ⊥ λ because ν lives on Q and λ lives on R \ Q.
The hard work for proving the next result has already been done in proving the
Hahn Decomposition Theorem (9.23).
9.30 Jordan Decomposition Theorem
• Every real measure is the difference of two finite (positive) measures that are
singular with respect to each other.
• More precisely, suppose ν is a real measure on a measurable space (X, S).
Then there exist unique finite (positive) measures ν + and ν − on (X, S) such
that
9.31 ν = ν + − ν − and ν + ⊥ ν − .
Furthermore,
|ν| = ν + + ν − .
Proof Let X = A ∪ B be a Hahn decomposition of ν as in 9.23. Define functions
ν + : S→[0, ∞) and ν − : S→[0, ∞) by
ν + (E) =ν(E ∩ A) and ν − (E) =−ν(E ∩ B).
The countable additivity of ν implies ν + and ν − are finite (positive) measures on
(X, S), with ν = ν + − ν − and ν + ⊥ ν − .
The definition of the total variation
measure and 9.31 imply that
Camille Jordan (1838–1922) is also
known for certain matrices that are
|ν| = ν + + ν − , as you should verify.
0 except along the diagonal and the
The equations ν = ν + − ν − and
line above it.
|ν| = ν + + ν − imply that
ν + = |ν| + ν
2
and
ν − = |ν|−ν .
2
Thus the finite (positive) measures ν + and ν − are uniquely determined by ν and the
conditions in 9.31.
Measure, Integration & Real Analysis, by Sheldon Axler