Math 447, Homework 3.∗
1.
Rb
(a) Let α be increasing. Show that if a f dα = 0 for all f ∈ C[a, b], then α is a constant
function (very easy; cf. Exercise 13).
Rb
(b) Give an example of a non-constant α ∈ BV [a, b] such that for all f ∈ C[a, b], a f dα = 0
(cf. Exercise 44).
(c) Let α ∈ BV0 [a, b] (that is, α(a) = 0 and α is right-continuous on (a, b)). Suppose that
Rb
f dα = 0 for all f ∈ C[a, b]. Show that for any two points of continuity c < d of α,
a
α(c) = α(d) (hint: choose the functions f to approximate χ[c,d] ; cf. Exercise 54). Conclude
that α = 0.
2. Let α ∈ BV [a, b]. Prove that the sum of the jumps
X
|α(x+) − α(x−)| ≤ Vab α,
a≤x≤b
and in particular the sum on the left-hand side is finite. Hint: first prove this for increasing α, and
then use the α = p − n form of the Jordan decomposition.
3. Exercise 14.41 (page 231).
4. Exercise 14.43 (page 231). Follow the hint, and also use Exercise 30.
Quiz 3.
(a) Exercise 14.52 (page 236). You may want to do Exercise 13.16 first. Hint: for the second
and third parts, you could use integration by parts and Problem 2.
(b) Exercise 14.53 (page 236) (easy given when you already proved).
∗
c 2016 by Michael Anshelevich.
1
Math 446, Honors Homework 3
1.
Rb
(a) Let α be increasing. Show that if a f dα = 0 for all f ∈ C[a, b], then α is a constant
function (very easy; cf. Exercise 13).
Rb
(b) Give an example of a non-constant α ∈ BV [a, b] such that for all f ∈ C[a, b], a f dα = 0
(cf. Exercise 44).
(c) Let α ∈ BV0 [a, b] (that is, α(a) = 0 and α is right-continuous on (a, b)). Suppose that
Rb
f dα = 0 for all f ∈ C[a, b]. Show that for any two points of continuity c < d of α,
a
α(c) = α(d) (hint: choose the functions f to approximate χ[c,d] ; cf. Exercise 54). Conclude
that α = 0.
2. Let α ∈ BV [a, b]. Prove that the sum of the jumps
X
|α(x+) − α(x−)| ≤ Vab α,
a≤x≤b
and in particular the sum on the left-hand side is finite. Hint: first prove this for increasing α, and
then use the α = p − n form of the Jordan decomposition.
3. Exercise 14.48 (page 232).
4. Exercise 14.43 (page 231). Follow the hint, and also use Exercise 30.
Honors Quiz 3.
(a) Exercise 14.52 (page 236). You may want to do Exercise 13.16 first. Hint: for the second
and third parts, you could use integration by parts and Problem 2.
(b) Exercise 14.54 (page 236).