Measure, Integration & Real Analysis, 2021a

86 Chapter 3 Integration
12 Show that there exists a Borel measurable function f : R → (0, ∞) such that
∫
χI f dλ = ∞ for every nonempty open interval I ⊂ R, where λ denotes
Lebesgue measure on R.
13 Give an example to show that the Monotone Convergence Theorem (3.11) can
fail if the hypothesis that f 1 , f 2 ,...are nonnegative functions is dropped.
14 Give an example to show that the Monotone Convergence Theorem can fail if
the hypothesis of an increasing sequence of functions is replaced by a hypothesis
of a decreasing sequence of functions.
[This exercise shows that the Monotone Convergence Theorem should be called
the Increasing Convergence Theorem. However, see Exercise 20.]
15 Suppose λ is Lebesgue measure on R and f : R → [−∞, ∞] is a Borel measurable
function such that ∫ f dλ is defined.
(a) For ∫ t ∈ R, define f t : R → [−∞, ∞] by f t (x) = f (x − t). Prove that
ft dλ = ∫ f dλ for all t ∈ R.
(b) For
∫
t ∈ R, define f t : R → [−∞, ∞] by f t (x) = f (tx). Prove that
ft dλ = 1 ∫
|t| f dλ for all t ∈ R \{0}.
16 Suppose S and T are σ-algebras on a set X and S ⊂ T . Suppose μ 1 is a
measure on (X, S), μ 2 is a measure on (X, T ), and μ 1 (E) =μ 2 (E) for all
E ∈S. Prove that if f : X → [0, ∞] is S-measurable, then ∫ f dμ 1 = ∫ f dμ 2 .
For x 1 , x 2 ,... a sequence in [−∞, ∞], define lim inf x k by
k→∞
lim inf x k = lim inf{x k, x k+1 ,...}.
k→∞ k→∞
Note that inf{x k , x k+1 ,...} is an increasing function of k; thus the limit above
on the right exists in [−∞, ∞].
17 Suppose that (X, S, μ) is a measure space and f 1 , f 2 ,...is a sequence of nonnegative
S-measurable functions on X. Define a function f : X → [0, ∞] by
f (x) =lim inf f k(x).
k→∞
(a) Show that f is an S-measurable function.
(b) Prove that
∫
∫
f dμ ≤ lim inf f k dμ.
k→∞
(c) Give an example showing that the inequality in (b) can be a strict inequality
even when μ(X) < ∞ and the family of functions { f k } k∈Z + is uniformly
bounded.
[The result in (b) is called Fatou’s Lemma. Some textbooks prove Fatou’s Lemma
and then use it to prove the Monotone Convergence Theorem. Here we are taking
the reverse approach—you should be able to use the Monotone Convergence
Theorem to give a clean proof of Fatou’s Lemma.]
Measure, Integration & Real Analysis, by Sheldon Axler