Measure, Integration & Real Analysis, 2021a

Section 3B Limits of Integrals & Integrals of Limits 91
Proof Suppose ε > 0. Let h : X → [0, ∞) be a simple S-measurable function such
that 0 ≤ h ≤ g and ∫ ∫
g dμ − h dμ < ε 2 ;
the existence of a function h with these properties follows from 3.9. Let
H = max{h(x) : x ∈ X}
and let δ > 0 be such that Hδ <
2 ε .
Suppose B ∈Sand μ(B) < δ. Then
∫ ∫
∫
g dμ = (g − h) dμ +
B
≤
B
∫
B
h dμ
(g − h) dμ + Hμ(B)
< ε 2 + Hδ
as desired.
< ε,
Some theorems, such as Egorov’s Theorem (2.85) have as a hypothesis that the
measure of the entire space is finite. The next result sometimes allows us to get
around this hypothesis by restricting attention to a key set of finite measure.
3.29 integrable functions live mostly on sets of finite measure
Suppose (X, S, μ) is a measure space, g : X → [0, ∞] is S-measurable, and
∫ g dμ < ∞. Then for every ε > 0, there exists E ∈Ssuch that μ(E) < ∞ and
∫
X\E
g dμ < ε.
Proof
3.30
Suppose ε > 0. Let P be an S-partition A 1 ,...,A m of X such that
∫
g dμ < ε + L(g, P).
Let E be the union of those A j such that inf g > 0. Then μ(E) < ∞ (because
A j
otherwise ∫ we would have L(g, P) = ∞, which contradicts the hypothesis that
g dμ < ∞). Now
∫ ∫ ∫
g dμ = g dμ − χ E
g dμ
X\E
< ( ε + L(g, P) ) −L(χ E
g, P)
= ε,
where the second line follows from 3.30 and the definition of the integral of a
nonnegative function, and the last line holds because inf g = 0 for each A j not
A
contained in E.
j
Measure, Integration & Real Analysis, by Sheldon Axler