# PROBLEM SET VII “THROUGH INTUITION” f t

```PROBLEM SET VII
“THROUGH INTUITION”
DUE FRIDAY,  OCTOBER 
Exercise . For each of the following functions f : [0, +∞) . R, ﬁnd the set
∫oft real numbers t ≥ 0 such that f is Riemann-integrable on [0, t], and compute
f:
0
{
N
∑
√
xk
1/(x − 1) if x ̸= 1,
f (x) = x;
f (x) =
f (x) =
.
k!
0
if x = 1;
k=0
(In the last example, N is any natural number.)
Deﬁnition. We say a subset E ⊂ R is bounded if it is a subset E ⊂ [a, b] of
a closed interval. In this case, deﬁne the indicator function of E as the function
χ E : [a, b] . R given by the formula
{
1 if x ∈ E;
χ E (x) :=
0 if x ∈
/ E.
We say that E is Jordan-measurable if the indicator function χ E is Riemann integrable on [a, b]. In this case, the Jordan measure of E is the value
∫ b
μ(E) :=
χ E.
a
Exercise . Show that a ﬁnite set E ⊂ R is Jordan-measurable, and that its Jordan measure is 0. Is it true that any bounded countable set is Jordan-measurable?
Exercise . Show that: () the union of two Jordan-measurable sets is Jordanmeasurable, () the intersection of two Jordan-measurable sets is Jordan-measurable,
and () the complement of a Jordan-measurable set is Jordan-measurable.
Exercise . Show that if E ⊂ R is a Jordan-measurable set, and if f : R .
map given by f (x) = ax + b for real numbers a and b, then the set
f (E) := {x ∈ R | there exists u ∈ E such that x = f (u)}
is also Jordan-measurable, and
μ( f (E)) = aμ(E).

R is a

DUE FRIDAY,  OCTOBER 
Exercise⋆ . Suppose f : [c, d] . R continuous and g : [a, b] . [c, d] Riemannintegrable. Prove that the composite f ◦ g : [a, b] . R is also Riemann-integrable.
Provide a counterexample that demonstrates that if we had assumed only that f
and g are Riemann-integrable, this statement would be false.
Deﬁnition. Suppose E ⊂ [a, b]. Now for any function f : [a, b] . R, we say that
f is Riemann-integrable on E if the product χ E &middot; f is Riemann integrable on [a, b],
and we write
∫
∫
b
f :=
E
a
χ E &middot; f.
Exercise⋆ . Show that if f, g : [a, b] . R are Riemann-integrable functions,
then the product f &middot; g : [a, b] . R is also Riemann-integrable. Deduce that if
E ⊂ [a, b] is Jordan measurable, and if f : [a, b] . R is Riemann integrable on
[a, b] then it is also Riemann-integable on E. [For the ﬁrst part, observe that 4f&middot;g =
(f + g)2 − (f − g)2 .]
Exercise . Give an example to show that the converse above is false. at is, ﬁnd
two bounded functions f, g : [a, b] . R, one of which is not Riemann integrable,
such that the product f &middot; g : [a, b] . R is Riemann-integrable.
Exercise . Suppose E a Jordan-measurable set, and suppose f a Riemann-integrable
function on E. Show that if A ⊂ E is a Jordan-measurable set such that μ(E − A)
is zero, then f is Riemann-integrable on A, and
∫
∫
f = f.
A
E
```