Measure, Integration & Real Analysis, 2021a

Section 5B Iterated Integrals 135
EXERCISES 5B
1 (a) Let λ denote Lebesgue measure on [0, 1]. Show that
and
∫
∫
[0, 1]
[0, 1]
∫
∫
[0, 1]
[0, 1]
x 2 − y 2
(x 2 + y 2 ) 2 dλ(y) dλ(x) =π 4
x 2 − y 2
(x 2 + y 2 ) 2 dλ(x) dλ(y) =− π 4 .
(b) Explain why (a) violates neither Tonelli’s Theorem nor Fubini’s Theorem.
2 (a) Give an example of a doubly indexed collection {x m,n : m, n ∈ Z + } of
real numbers such that
∞
∑
m=1
∞
∑ x m,n = 0
n=1
and
∞
∑
n=1
∞
∑ x m,n = ∞.
m=1
(b) Explain why (a) violates neither Tonelli’s Theorem nor Fubini’s Theorem.
3 Suppose (X, S) is a measurable space and f : X → [0, ∞] is a function. Let B
denote the σ-algebra of Borel subsets of (0, ∞). Prove that U f ∈S⊗Bif and
only if f is an S-measurable function.
4 Suppose (X, S) is a measurable space and f : X → R is a function. Let
graph( f ) ⊂ X × R denote the graph of f :
graph( f )={ ( x, f (x) ) : x ∈ X}.
Let B denote the σ-algebra of Borel subsets of R. Prove that graph( f ) ∈S⊗B
if f is an S-measurable function.
Measure, Integration & Real Analysis, by Sheldon Axler