Measure, Integration & Real Analysis, 2021a

Section 3B Limits of Integrals & Integrals of Limits 99
EXERCISES 3B
1 Give an example of a sequence f 1 , f 2 ,...of functions from Z + to [0, ∞) such
that
lim f k(m) =0
k→∞
∫
for every m ∈ Z + but lim f k dμ = 1, where μ is counting measure on Z + .
k→∞
2 Give an example of a sequence f 1 , f 2 ,... of continuous functions from R to
[0, 1] such that
lim f k(x) =0
k→∞
∫
for every x ∈ R but lim f k dλ = ∞, where λ is Lebesgue measure on R.
k→∞
3 Suppose λ is Lebesgue measure on R and f : R → R is a Borel measurable
function such that ∫ | f | dλ < ∞. Define g : R → R by
∫
g(x) = f dλ.
(−∞, x)
Prove that g is uniformly continuous on R.
4 (a) Suppose (X, S, μ) is a measure space with μ(X) < ∞. Suppose that
f : X → [0, ∞) is a bounded S-measurable function. Prove that
∫
{ m }
f dμ = inf ∑ μ(A j ) sup f : A 1 ,...,A m is an S-partition of X .
j=1 A j
(b) Show that the conclusion of part (a) can fail if the hypothesis that f is
bounded is replaced by the hypothesis that ∫ f dμ < ∞.
(c) Show that the conclusion of part (a) can fail if the condition that μ(X) < ∞
is deleted.
[Part (a) of this exercise shows that if we had defined an upper Lebesgue sum,
then we could have used it to define the integral. However, parts (b) and (c) show
that the hypotheses that f is bounded and that μ(X) < ∞ would be needed if
defining the integral via the equation above. The definition of the integral via the
lower Lebesgue sum does not require these hypotheses, showing the advantage
of using the approach via the lower Lebesgue sum.]
5 Let λ denote Lebesgue measure on R. Suppose f : R → R is a Borel measurable
function such that ∫ | f | dλ < ∞. Prove that
∫
∫[−k, f dλ = f dλ.
k]
lim
k→∞
6 Let λ denote Lebesgue measure on R. Give an example of a continuous function
f : [0, ∞) → R such that lim t→∞
∫[0, t] f dλ exists (in R)but∫ [0, ∞)
f dλ is not
defined.
Measure, Integration & Real Analysis, by Sheldon Axler