Measure, Integration & Real Analysis, 2021a

126 Chapter 5 Product Measures
5.23 Definition iterated integrals
Suppose (X, S, μ) and (Y, T , ν) are measure spaces and f : X × Y → R is a
function. Then
∫ ∫
∫ (∫
)
f (x, y) dν(y) dμ(x) means f (x, y) dν(y) dμ(x).
X
Y
In other words, to compute ∫ ∫
X Y
f (x, y) dν(y) dμ(x), first (temporarily) fix x ∈
X and compute ∫ Y
f (x, y) dν(y) [if this integral makes sense]. Then compute the
integral with respect to μ of the function x ↦→ ∫ Y
f (x, y) dν(y) [if this integral
makes sense].
X
Y
5.24 Example iterated integrals
If λ is Lebesgue measure on [0, 4], then
∫ ∫
∫
[0, 4] (x2 + y) dλ(y) dλ(x) =
[0, 4] (4x2 + 8) dλ(x)
[0, 4]
and
∫
[0, 4]
= 352
3
∫
∫
[0, 4] (x2 + y) dλ(x) dλ(y) =
[0, 4]
( 64
3 + 4y )
dλ(y)
= 352
3 .
The two iterated integrals in this example turned out to both equal 352
3
, even though
they do not look alike in the intermediate step of the evaluation. As we will see in the
next section, this equality of integrals when changing the order of integration is not a
coincidence.
The definition of (μ × ν)(E) given below makes sense because the inner integral
below equals ν([E] x ), which makes sense by 5.6 (or use 5.9), and then the outer
integral makes sense by 5.20(a).
The restriction in the definition below to σ-finite measures is not bothersome because
the main results we seek are not valid without this hypothesis (see Example 5.30
in the next section).
5.25 Definition product of two measures; μ × ν
Suppose (X, S, μ) and (Y, T , ν) are σ-finite measure spaces. For E ∈S⊗T,
define (μ × ν)(E) by
∫ ∫
(μ × ν)(E) = χ E
(x, y) dν(y) dμ(x).
X
Y
Measure, Integration & Real Analysis, by Sheldon Axler