Measure, Integration & Real Analysis, 2021a

Section 9B Decomposition Theorems 273
Similarly, if f equals the characteristic function of {x ∈ X : g(x) < 0}, then the
left side of 9.39 is greater than or equal to 0 and the right side of 9.39 is less than
or equal to 0; hence both sides are 0. Thus ∫ fgdμ = 0, which implies (with this
choice of f ) that μ({x ∈ X : g(x) < 0}) =0.
Because ν ≪ μ, the two previous paragraphs imply that
ν({x ∈ X : g(x) ≥ 1}) =0 and ν({x ∈ X : g(x) < 0}) =0.
Thus we can modify g (for example by redefining g to be 1 2
on the two sets appearing
above; both those sets have ν-measure 0 and μ-measure 0) and from now on we can
assume that 0 ≤ g(x) < 1 for all x ∈ X and that 9.39 holds for all f ∈L 2 (ν + μ).
Hence we can define h : X → [0, ∞) by
h(x) =
g(x)
1 − g(x) .
Suppose E ∈S. For each k ∈ Z + , let
⎧
⎨ χ (x) E
if χ E (x)
f k (x) =
1−g(x) 1−g(x) ≤ k,
⎩
0 otherwise.
Then f k ∈L 2 (ν + μ). Now9.39 implies
∫
∫
f k (1 − g) dν =
Taking f = χ E
/(1 − g) in 9.39
would give ν(E) = ∫ E
h dμ, but this
function f might not be in
L 2 (ν + μ) and thus we need to be a
bit more careful.
f k g dμ.
Taking the limit as k → ∞ and using the Monotone Convergence Theorem (3.11)
shows that
∫ ∫
9.40
1 dν = h dμ.
E
Thus dν = h dμ, completing the proof in the case where both ν and μ are (positive)
finite measures [note that h ∈L 1 (μ) because h is a nonnegative function and we can
take E = X in the equation above].
Now relax the assumption on μ to the hypothesis that μ is a σ-finite measure.
Thus there exists an increasing sequence X 1 ⊂ X 2 ⊂···of sets in S such that
⋃ ∞k=1
X k = X and μ(X k ) < ∞ for each k ∈ Z + .Fork ∈ Z + , let ν k and μ k denote
the restrictions of ν and μ to the σ-algebra on X k consisting of those sets in S that
are subsets of X k . Then ν k ≪ μ k . Thus by the case we have already proved, there
exists a nonnegative function h k ∈L 1 (μ k ) such that dν k = h k dμ k .Ifj < k, then
∫
∫
h j dμ = ν(E) = h k dμ
E
for every set E ∈Swith E ⊂ X j ; thus μ({x ∈ X j : h j (x) ̸= h k (x)}) =0. Hence
there exists an S-measurable function h : X → [0, ∞) such that
μ ( {x ∈ X k : h(x) ̸= h k (x)} ) = 0
for every k ∈ Z + . The Monotone Convergence Theorem (3.11) can now be used to
show that 9.40 holds for every E ∈S. Thus dν = h dμ, completing the proof in the
case where ν is a (positive) finite measure.
Measure, Integration & Real Analysis, by Sheldon Axler
E
E