Measure, Integration & Real Analysis, 2021a

Section 3B Limits of Integrals & Integrals of Limits 95
We previously defined the notation ∫ b
a
f to mean the Riemann integral of f .
Because the Riemann integral and Lebesgue integral agree for Riemann integrable
functions (see 3.34), we now redefine ∫ b
a
f to denote the Lebesgue integral.
3.39 Definition ∫ b
a
f
Suppose −∞ ≤ a < b ≤ ∞ and f : (a, b) → R is Lebesgue measurable. Then
• ∫ b
a
f and ∫ b
a f (x) dx mean ∫ (a,b)
f dλ, where λ is Lebesgue measure on R;
• ∫ a
b f is defined to be − ∫ b
a f .
The definition in the second bullet point above is made so that equations such as
∫ b
a
f =
∫ c
remain valid even if, for example, a < b < c.
Approximation by Nice Functions
a
In the next definition, the notation ‖ f ‖ 1 should be ‖ f ‖ 1,μ because it depends upon
the measure μ as well as upon f . However, μ is usually clear from the context. In
some books, you may see the notation L 1 (X, S, μ) instead of L 1 (μ).
f +
∫ b
c
f
3.40 Definition ‖ f ‖ 1 ; L 1 (μ)
Suppose (X, S, μ) is a measure space. If f : X → [−∞, ∞] is S-measurable,
then the L 1 -norm of f is denoted by ‖ f ‖ 1 and is defined by
∫
‖ f ‖ 1 = | f | dμ.
The Lebesgue space L 1 (μ) is defined by
L 1 (μ) ={ f : f is an S-measurable function from X to R and ‖ f ‖ 1 < ∞}.
The terminology and notation used above are convenient even though ‖·‖ 1 might
not be a genuine norm (to be defined in Chapter 6).
3.41 Example L 1 (μ) functions that take on only finitely many values
Suppose (X, S, μ) is a measure space and E 1 ,...,E n are disjoint subsets of X.
Suppose a 1 ,...,a n are distinct nonzero real numbers. Then
a 1 χ
E1
+ ···+ a n χ
En
∈L 1 (μ)
if and only if E k ∈Sand μ(E k ) < ∞ for all k ∈{1, . . . , n}. Furthermore,
‖a 1 χ
E1
+ ···+ a n χ
En
‖ 1 = |a 1 |μ(E 1 )+···+ |a n |μ(E n ).
Measure, Integration & Real Analysis, by Sheldon Axler