EXERCISES AND PROBLEMS FOR CHAPTER 2:

MEASURES

A. Problems and Exercises for everyone:

All problems and exercises in parts B and C.

B. Non-assessed Problems and Exercises (corrected in class):

0.1.1; 0.1.3; 0.1.5; 0.1.7 (a), (b); 0.2.1; 0.2.2; 0.2.3; 0.2.4;
0.2.9; 0.2.11; 0.2.13; 0.3.1; 0.3.3; 0.4.2; 0.5.4, 0.5.6.

C. Assessed Assignments (to be submitted):

0.1.2; 0.1.4; 0.1.8 (a), (b); 0.2.1; 0.2.6; 0.2.8; 0.2.12; ??;
0.3.2; 0.3.8; 0.5.3; 0.5.5.

D. Bonus Problems and Exercises: Remaining exercises and problems.

0.1 ALGEBRAS AND σ-ALGEBRAS

Exercise 0.1.1. Show that a nonempty family A ⊂ P(X) is an algebra
provided that for all A, B ∈ A we have Ac ∈ A and A ∩ B ∈ A.

Exercise 0.1.2. Prove that for
any class E of
sets in X and any mapping
f : X → X, one has σ f −1 (E) = f −1 σ(E) , where f −1 (E) = {f −1 (E) :
E ∈ E}.

Exercise 0.1.3. Prove that every countable set in R is a Borel set.

Exercise 0.1.4. If Y is a nonempty Borel subset of R, show that the Borel

algebra of the subspace Y is {A ∈ B(R) : A ⊂ Y }.

Exercise 0.1.5. An Fσ -set is any countable union of closed sets, and a Gδ -

set is any countable intersection of open sets. Prove that both types of sets
are Borel sets.

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Exercise 0.1.6. Let {En } be a sequence in an algebra A, thenSthere is a
sequence {Fn } of disjoint sets of A such that Fn ⊂ En for each n, kn=1 Bn =
Sk S∞ S∞
n=1 An for each n, and n=1 Bn = n=1 An .

Exercise 0.1.7. Prove that B(R) is generated by each of the following:

(a) the open intervals E1 = {(a, b) : a < b}, a, b ∈ R;

(b) the closed intervals E2 = {[a, b] : a < b}, a, b ∈ R;

(c) the half-open intervals E3 = {(a, b] : a < b} or E4 = {[a, b) : a < b}

(a, b ∈ R);

(d) the open rays E5 = {(a, ∞) : a ∈ R} or E6 = {(−∞, b) : b ∈ R};

(e) the open rays E7 = {[a, ∞) : a ∈ R} or E8 = {(−∞, b] : b ∈ R}.

Exercise 0.1.8. Let D be an arbitrary dense set in R (say D = Q). Prove

that B(R) is generated by any of the following classes of sets:

(a) the open intervals F1 = {(a, b) : a < b}, a, b ∈ D;

(b) the closed intervals F2 = {[a, b] : a < b}, a, b ∈ D;

(c) the half-open intervals F3 = {(a, b] : a < b} or F4 = {[a, b) : a < b},

a, b ∈ D;

(d) the open rays F5 = {(a, ∞) : a ∈ D} or F6 = {(∞, b) : b ∈ D};

(e) the open rays F7 = {[a, ∞) : a ∈ D} or F8 = {(∞, b] : b ∈ D}.

0.2 MEASURES
Exercise 0.2.1. Let (X, M, µ) be a measure space. Show that if µ is σ-
S for every set E ∈ M, there exists a sequence {En } ⊂ M such
finite, then
that E = n En and µ(En ) < ∞ for each n, i.e., every E ∈ M is σ-finite.

Exercise 0.2.2. Show that a countable union of null sets is again a null set.

2
Exercise 0.2.3. Let (X, M, µ) be a measure space, and let {Ai }∞
i=1 ⊂ M.
Prove that [∞  [ n 
µ Ai = lim µ Ai .
n→∞
i=1 i=1

Exercise 0.2.4. Let M be a σ-algebra of subsets of a set X and the set

function µ : M → [0, ∞) be finitely additive.
(a) Prove that µ is a measure if and only if whenever {An } ⊂ M, A1 ⊂
A2 ⊂ · · · , then
[∞ 
µ An = lim µ(An ).
n→∞
n=1

(b) Suppose that µ is finite. Prove that µ is

T a measure if and only if
whenever {An } ⊂ M, A1 ⊃ A2 ⊃ · · · and ∞ n=1 An = ∅, then

lim µ(An ) = 0.
n→∞

Exercise 0.2.5. Let A be the algebra of sets A ⊂ N such that either A or

N \ A is finite. For finite A, let µ(A) = 0, and for A with a finite complement
let µ(A) = 1. Then µ is an additive, but not countably additive set function.

Exercise 0.2.6. Let X be a countably infinite set, and let A be the algebra
consisting of all finite subsets of X and their complements. If A is finite, set
µ(A) = 0, and if Ac is finite, set µ(A) = 1.
(a) Show that µ is finitely additive but not countably additive on A.

(b) Show that X is the limit of a sequence of sets An ∈ A, A1 ⊂ A2 ⊂ · · ·

such that µ(An ) = 0 for all n but µ(X) = 1.

Exercise 0.2.7. Let µ be counting measure on X, where X is an infinite

T∞
set. Show that there is a sequence of sets A1 ⊃ A2 ⊃ · · · with n=1 An = ∅
and limn→∞ µ(An ) 6= 0.

Exercise 0.2.8. Let µ1 , . . . , µn be measures on (X, M) and c1 , . . . , cn posi-

tive numbers. Show that µ := c1 µ1 + · · · + cn µn is a measure on (X, M).

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Exercise 0.2.9. Let (X, M, µ) be a measure space. Prove that for A, B ∈
M,
µ(A ∪ B) + µ(A ∩ B) = µ(A) + µ(B). (0.2.1)
Applications: Show that if µ is a probability measure, then for any mea-
surable sets A, B we have
(i) µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B), and

(ii) min{µ(A), µ(B)} ≥ µ(A ∩ B) ≥ µ(A) + µ(B) − 1.

Exercise 0.2.10. Given a measure space (X, M, µ) and E ∈ M, define

µE (A) = µ(A ∩ E) for A ∈ M. Show that µE is a measure on M.

Exercise 0.2.11. Let (X, M, P ) be a probability space and B ∈ M with

P (B) > 0. The number

P (A ∩ B)
P (A|B) =
P (B)
is called the conditional probability of A given B.
Show that the function A 7→ P (A|B) is a probability measure on the
σ-algebra M.

Exercise 0.2.12. Given a probability space (X, M, P ) we say that the ele-
ments of M are events. The events A, B are independent if

P (A ∩ B) = P (A) · P (B).

Show that if A and B are independent events, then Ac and B are also inde-
pendent.

Exercise 0.2.13. The symmetric difference of two sets A and B is A∆B =

(A \ B) ∪ (B \ A). Let (X, A, µ) be a measure space.
(a) Show that if A and B are measurable and µ(A∆B) = 0, then µ(A) =
µ(B).

(b) Show that if µ is complete, A ∈ A and µ(A∆B) = 0, then B ∈ A.

Exercise 0.2.14. Let (X, M) be a measurable space. Verify the following:

4
 (a) If µ and µ are measures defined on M, then the set function λ defined
on M by λ(E) = µ(E) + ν(E) also is a measure. We denote λ by µ + ν.

(b) If µ and ν are measures on M and µ ≥ ν, then there is a measure ξ on

M for which µ = ν + ξ.

(c) If ν is σ-finite, the measure ξ in (b) is unique.

(d) Show that in general the measure ξ in (b) need not be unique but that
there is always a smallest such ξ.

0.3 OUTER MEASURES

Exercise 0.3.1. Let X = {a, b} and define µ∗ (∅) = 0, µ∗ ({a}) = 1, µ∗ ({b}) =
2, and µ∗ (X) = 2. Show that µ∗ is an outer measure but is not additive.

Exercise 0.3.2. Let X be any set. Define ν : P(X) → [0, ∞] by defining

ν(∅) = 0 and for E ⊂ X, E 6= ∅, defining ν(E) = ∞. Show that ν is an
outer measure.

Exercise 0.3.3. Prove that for any outer measure µ∗ and any set A such
that µ∗ (A) = 0, A is µ∗ -measurable.

Exercise 0.3.4. Let X = N and E be the family of all singletons and the
whole set N. Let µ(∅) = 0, µ({n}) = 21n , and µ(N) = 2. Determine µ∗ (N)
and all µ∗ -measurable sets.

Exercise 0.3.5. Prove that if µ∗ is an outer measure on X and if B ⊂ X,

µ∗ (B) = 0, then µ∗ (A ∪ B) = µ∗ (A \ B) = µ∗ (A).

Exercise 0.3.6. Let µ∗ be an outer measure on X, and let Y ⊂ X. Define

ν ∗ (A) = µ∗ (A) when A ⊂ Y . Is ν ∗ an outer measure on Y ?

Exercise 0.3.7. Let µ∗ be an outer measure on X, and let Y ⊂ X. Define

ν ∗ (A) = µ∗ (Y ∩ A). Is ν ∗ an outer measure on X?

Exercise 0.3.8. Show that a subset E of X is µ∗ -measurable if and only

if for each  > 0 there exists a measurable set F such that F ⊂ E and
µ(E \ F ) < .

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0.4 THE LEBESGUE MEASURE ON Rn
Exercise 0.4.1. Let I1 , I2 , . .P
. , In be a finite set of intervals covering the
rationals in [0, 1]. Show that nk=1 m(Ik ) ≥ 1.

Exercise 0.4.2. Let S be a subset of Rn such that for each  > 0 there is
a closed set F contained in S for which m∗ (S \ F ) < . Prove that S is
Lebesgue measurable.

Exercise 0.4.3. Prove that a subset E of Rn is Lebesgue measurable if for

each  > 0, there exists an open set U such that E ⊂ U and m∗ (U \ E) < .

Exercise 0.4.4. Let {Ak } beSan increasing sequence of subsets of Rn , that is,
A1 ⊂ A2 ⊂ · · · , and let A = ∞ ∗ ∗
k=1 Ak . Show that limk→∞ m (Ak ) = m (A).
(Hint. Let Bk be a Lebesgue measurable Tk∞⊂ Bk and m(Bk ) =
set with A
m∗ (Ak ), k = 1, 2, ... Set Cm = ∞
S
k=m kB and C = m=1 Cm . Show that
∗ ∗
C ⊃ A, m (Ak ) = m(Bk ) = m(Ck ), and limk→∞ m (Ak ) = m(C).)

0.5 BOREL MEASURES ON R

Exercise 0.5.1. Show that if f : [a, b] → [c, d] is both monotone and onto,
then f is continuous.

Exercise 0.5.2. Show that any monotone function f : R → R has points of

continuity in every (nonempty) open interval.

Exercise 0.5.3. Show that a strictly increasing function that is defined

on an interval is Lebesgue measurable and then use this to show that a
monotone function that is defined on an interval is Lebesgue measurable.
(Every monotone function is measurable.)

A distribution function on R is a function F : R → R that is increasing

and right continuous.

Exercise 0.5.4. If F is a distribution function, the measure µF (I) of any

interval I may be expressed in terms of F : for −∞ < a < b < ∞,



µF (a, b] = F (b) − F (a), µF [a, b] = F (b) − F (a−)



µF (a, b) = F (b−) − F (a), µF [a, b) = F (b−) − F (a−).

6
Thus if F is continuous at a and b, all four expressions are equal. Show that
F is continuous if and only if µF ({y}) = 0 for all y.

Exercise 0.5.5. Let F be the distribution function on R given by



 0 if x < −1;

1 + x if − 1 ≤ x < 0;
F (x) =


 2 + x2 if 0 ≤ x < 2;
if x ≥ 2.

9

If µ is the Lebesgue-Stieltjes measure corresponding to F , compute the mea-

sure of each of the following sets:
(a) {2}, (d) [0, 12 ) ∪ (1, 2],
1
(b) [− 2 , 3) (e) {x : |x| + 2x2 > 1}.
(c) (−1, 0] ∪ (1, 2),
(Hint: Apply Exercise 0.5.4.)

Exercise 0.5.6. A probability distribution is by definition a probability

measure P on R defined on the σ-algebra of Borel sets B(R). The function
F : R → [0, 1] defined as


F (x) = P (−∞, x] , x ∈ R,

is called the (cumulative) distribution function. Prove the following

properties of F .

(a) F (x) ≤ F (y) for every x ≤ y (that is, F is non-decreasing);

(b) limx→a F (x) = F (a) for each a ∈ R (that is, F is right-continuous);

(c) limx→−∞ F (x) = 0.

(d) limx→+∞ F (x) = 1.

Exercise 0.5.7. Show that if F = χ[c,∞) , then mF = δc , the Dirac measure

concentrated at c.

Exercise 0.5.8. Determine the probability measure on B(R) which has

f (x) = max{0, min{x, 1}} as its distribution function.