Measure, Integration & Real Analysis, 2021a

82 Chapter 3 Integration
The condition ∫ | f | dμ < ∞ is equivalent to the condition ∫ f + dμ < ∞ and
∫ f − dμ < ∞ (because | f | = f + + f − ).
3.19 Example a function whose integral is not defined
Suppose λ is Lebesgue measure on R and f : R → R is the function defined by
{
1 if x ≥ 0,
f (x) =
−1 if x < 0.
Then ∫ f dλ is not defined because ∫ f + dλ = ∞ and ∫ f − dλ = ∞.
The next result says that the integral of a number times a function is exactly what
we expect.
3.20 integration is homogeneous
Suppose (X, S, μ) is a measure space and f : X → [−∞, ∞] is a function such
that ∫ f dμ is defined. If c ∈ R, then
∫
∫
cf dμ = c
f dμ.
Proof First consider the case where f is a nonnegative function and c ≥ 0. IfP is
an S-partition of X, then clearly L(cf, P) =cL( f , P). Thus ∫ cf dμ = c ∫ f dμ.
Now consider the general case where f takes values in [−∞, ∞]. Suppose c ≥ 0.
Then
∫ ∫
∫
cf dμ = (cf) + dμ − (cf) − dμ
∫
=
(∫
= c
∫
= c
∫
cf + dμ −
∫
f + dμ −
f dμ,
cf − dμ
)
f − dμ
where the third line follows from the first paragraph of this proof.
Finally, now suppose c < 0 (still assuming that f takes values in [−∞, ∞]). Then
−c > 0 and
∫ ∫
∫
cf dμ = (cf) + dμ − (cf) − dμ
completing the proof.
∫
∫
= (−c) f − dμ − (−c) f + dμ
(∫
=(−c)
∫
= c
f dμ,
∫
f − dμ −
)
f + dμ
Measure, Integration & Real Analysis, by Sheldon Axler