MATH 321 - HOMEWORK #1
Due Friday, Jan 15.
P ROBLEM 1. Recall that we defined the Riemann sum
n
X
RS(P, {ti }, f, α) =
f(ti )∆αi .
i=1
This sum depends on the partition P, the tagging ti ∈ [xi−1 , xi ], the functions f and α. We
also defined the mesh of a partition P to be
kPk = max(xi − xi−1 ).
i
Let f, α : [0, 2] → R be as in Homework 1, Problem 2:
0 if x ≤ 1,
0
f(x) =
α(x) =
1 otherwise,
1
if x < 1,
otherwise.
Find a sequence of partitions P1 , P2 , P3 , . . ., with limk→∞ kPk k = 0, and two different
taggings for each partition {ti } and {si }, such that
lim RS(Pk , {ti }, f, α) = 1,
lim RS(Pk , {si }, f, α) = 0.
k→∞
k→∞
This problem shows that, even though f ∈ R(α), we cannot compute the integral by
arbitrary Riemann sums.
P ROBLEM 2. Let f, α : [0, 1] → R, with f bounded and α nondecreasing. Assume that f is
continuous except at the points in S = {0, 12 , 13 , 14 , . . .} and that α is continuous at all points
of S. Prove that then f ∈ R(α).
(Hint: Divide the domain [0, 1] into two intervals [0, δ] ∪ [δ, 1] for some small δ and make
the difference between the upper and lower sums small on each interval separately.)
P ROBLEM 3. Let f, α : [a, b] → R be nondecreasing functions. The goal of this problem is
to prove the integration by parts formula for Riemann-Stieltjes integrals:
Zb
Zb
fdα + αdf = f(b)α(b) − f(a)α(a).
(∗)
a
a
This formula should be read as: if one integral exists, then so does the other and then the
equality holds.
(1)Prove that for any partition P,
U(P, f, α) + L(P, α, f) = f(b)α(b) − f(a)α(a).
1
(Then also, interchanging f and α,
L(P, f, α) + U(P, α, f) = f(b)α(b) − f(a)α(a).)
(2)Show that if f ∈ R(α), then α ∈ R(f) and the equality (∗) holds.
P ROBLEM 4. Let f, α : [a, ∞) → R and assume that f ∈ R(α) on any finite interval [a, b],
a ≤ b. Define the improper integral
Z∞
Zb
fdα = lim fdα
b→∞ a
a
if this limit exists. Assume that the improper integral exists and prove that it can be
computed using upper and lower sums of partitions of [a, ∞).
More precisely, define a partition P of [a, ∞) to be a sequence {a = x0 < x1 < x2 < . . .},
where limi→∞ xi = ∞. For each partition define the upper and lower sums the same way
as before. These sums may be infinite. Define the upper and lower integrals as before,
using infimum and supremum over all partitions of [a, ∞). Prove that if f is integrable on
[a, ∞) according to the limit definition above, then
Z∞
Z∞
Z∞
fdα.
fdα =
fdα =
a
a
a
(Hint: Divide [a, ∞) into a sequence of finite intervals, for example Ik = [a + k, a + k + 1],
k = 0, 1, . . .. It should follow from what we prove in class that the improper integral is the
sum of integrals over these finite intervals. Now construct a partition that respects the
subdivision into Ik , where the difference between the upper and lower sum is small on
each Ik , and moreover, these differences sum up to ε. The same method can be used to
show that the upper and lower sums are finite for some P.)