Measure, Integration & Real Analysis, 2021a

90 Chapter 3 Integration
3.27 Definition almost every
Suppose (X, S, μ) is a measure space. A set E ∈Sis said to contain μ-almost
every element of X if μ(X \ E) =0. If the measure μ is clear from the context,
then the phrase almost every can be used (abbreviated by some authors to a. e.).
For example, almost every real number is irrational (with respect to the usual
Lebesgue measure on R) because |Q| = 0.
Theorems about integrals can almost always be relaxed so that the hypotheses
apply only almost everywhere instead of everywhere. For example, consider the
Bounded Convergence Theorem (3.26), one of whose hypotheses is that
lim f k(x) = f (x)
k→∞
for all x ∈ X. Suppose that the hypotheses of the Bounded Convergence Theorem
hold except that the equation above holds only almost everywhere, meaning there
is a set E ∈Ssuch that μ(X \ E) =0 and the equation above holds for all x ∈ E.
Define new functions g 1 , g 2 ,...and g by
{
f
g k (x) = k (x) if x ∈ E,
0 if x ∈ X \ E
and g(x) =
{
f (x) if x ∈ E,
0 if x ∈ X \ E.
Then
lim g k(x) =g(x)
k→∞
for all x ∈ X. Hence the Bounded Convergence Theorem implies that
which immediately implies that
lim
k→∞
lim
k→∞
∫
∫
∫
g k dμ =
∫
f k dμ =
because ∫ g k dμ = ∫ f k dμ and ∫ g dμ = ∫ f dμ.
Dominated Convergence Theorem
g dμ,
f dμ
The next result tells us that if a nonnegative function has a finite integral, then its
integral over all small sets (in the sense of measure) is small.
3.28 integrals on small sets are small
Suppose (X, S, μ) is a measure space, g : X → [0, ∞] is S-measurable, and
∫ g dμ < ∞. Then for every ε > 0, there exists δ > 0 such that
for every set B ∈Ssuch that μ(B) < δ.
∫
B
g dμ < ε
Measure, Integration & Real Analysis, by Sheldon Axler