Measure, Integration & Real Analysis, 2021a

132 Chapter 5 Product Measures
As you will see in the proof of Fubini’s Theorem, the function in 5.32(a) is defined
only for almost every x ∈ X and the function in 5.32(b) is defined only for almost
every y ∈ Y. For convenience, you can think of these functions as equaling 0 on the
sets of measure 0 on which they are otherwise undefined.
5.32 Fubini’s Theorem
Suppose (X, S, μ) and (Y, T , ν) are σ-finite measure spaces. Suppose
f : X × Y → [−∞, ∞] is S⊗T-measurable and ∫ X×Y
| f | d(μ × ν) < ∞.
Then ∫
| f (x, y)| dν(y) < ∞ for almost every x ∈ X
and
Y
∫
X
Furthermore,
∫
(a) x ↦→
(b)
and
∫
X×Y
y ↦→
Y
∫
X
∫
f d(μ × ν) =
| f (x, y)| dμ(x) < ∞ for almost every y ∈ Y.
f (x, y) dν(y) is an S-measurable function on X,
f (x, y) dμ(x) is a T -measurable function on Y,
X
∫
Y
∫
f (x, y) dν(y) dμ(x) =
Y
∫
X
f (x, y) dμ(x) dν(y).
Proof Tonelli’s Theorem (5.28) applied to the nonnegative function | f | implies that
x ↦→ ∫ Y
| f (x, y)| dν(y) is an S-measurable function on X. Hence
{ ∫
}
x ∈ X : | f (x, y)| dν(y) =∞ ∈S.
Y
Tonelli’s Theorem applied to | f | also tells us that
∫ ∫
| f (x, y)| dν(y) dμ(x) < ∞
X
Y
because the iterated integral above equals ∫ X×Y
| f | d(μ × ν). The inequality above
implies that
({ ∫
})
μ x ∈ X : | f (x, y)| dν(y) =∞ = 0.
Y
Recall that f + and f − are nonnegative S⊗T-measurable functions such that
| f | = f + + f − and f = f + − f − (see 3.17). Applying Tonelli’s Theorem to f +
and f − , we see that
∫
5.33 x ↦→
Y
∫
f + (x, y) dν(y) and x ↦→
Y
f − (x, y) dν(y)
Measure, Integration & Real Analysis, by Sheldon Axler