Contents
1. Set Functions: Algebra of Sets and Limits
1.1. Revision on Sets and Functions
2. Measurable spaces
2.1. Preliminaries
2.2. Algebra of Sets: Rings, algebras and σ−Algebras
2.3. Set Function
2.4. Measurable Spaces and Measure Spaces
2.5. Properties of Lebesgue Measure
3. The Lebesgue Integral
3.1. Measurable Functions
3.2. Integrals
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4
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5
6
7
11
12
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16
1. Set Functions: Algebra of Sets and Limits
NB:
• This chapter is a revision on sets and functions. However, if you encounter some
new concepts feel free to inform me that you are new to the concept.
• Whenever we use set theoretic conepts we assume that there is a universal set
X which contains all the sets under consderation, that should be clear from the
context.
1.1. Revision on Sets and Functions.
1.1.1. Sets.
Definition 1.1.
Subset: Let A and B be sets. We say A is a subset of B, A ⊂ B if and only if
x ∈ A =⇒ x ∈ B.
Equal sets: If A ⊂ B and B ⊂ A then A = B.
Singleton Set: A set with only one element, such as {x}, is called a singleton or
unit set.
Empty Set: A set with no elements is called an empty set.
Power set: Power set of set X, denoted by P (X), is the set of all subsets of X.
1.1.2. Operations on Set: Union, intersection and complement. The student is expected
to revise his/her knowledge on the following set operations.
Definition 1.2. Let Xbe a set.
Intersection: For A, B ∈ P (X) the intersection of A and B, denoted by A ∩ B is
defined as
^
A ∩ B := {x ∈ X : x ∈ A x ∈ B}.
1
2
Union: For A, B ∈ P (X) the union of A and B, denoted by A ∩ B is defined as
A ∪ B := {x ∈ X : x ∈ A ∨ x ∈ B}.
Complement: For A ∈ P (A) the complement of A (relative to X), denoted by Ac
or A0 , is defined by
Ac := {x ∈ X : x 6∈ A}.
Diffence: For A, B ∈ P (X) the difference of A and B or relative compliment of A
in B, denoted by A − B or A\B, is the set of all elements in A which are not in
B; i.e.,
A − B = {a ∈ A : a 6∈ B}
Symetric difference: For A, B ∈ P (X), the symmetric difference of A and B,
denoted by A∆B, is defined as
A∆B := (A − B) ∪ (B − A).
(Show that A∆B = (A ∪ B) − (A ∩ B))
Exercise 1.3. Fix set X as a universal set. Prove that for A, B, C ∈ P (X)
a) A ∪ B = B ∪ A
b) A ∩ B = B ∩ A
c) A∆B = B∆A
d) (A ∪ B) ∪ C = A ∪ (B ∪ C)
e) (A ∩ B) ∩ C = A ∩ (B ∩ C)
f ) A ∩ B = A ⇐⇒ A ⊆ B
g) A ∪ B = A ⇐⇒ B ⊆ B
h) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
j) A ∪ ∅ = A
k) A ∩ ∅ = ∅
l) A ∪ X = X
m) A ∩ X = A
n) A ⊂ B ⇐⇒ B c ⊂ Ac
o) ∅c = X
p) X c = ∅
1.1.3. Functions.
Definition 1.4. Some fundamental definitions:
• A function f from (or on) a set Xto (or into) a set Y we mean a rule which assigns
to ech x ∈ X a unique element f (x) ∈ Y .
• The set
{(x, f (x)) : x ∈ X} ⊂ X × Y,
is called the graph of the function f.
• A function f on X into Y is denoted as f : X → Y. The set X is called the domain
of f and Y is called the co-domain of f. We write D(f ) for domain of f.
• The range of f , denoted by R(f ) is given by the set {y ∈ Y : y = f (x) ∃ x ∈ X}.
• If R(f ) = Y then f is said to be an onto function.
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• Let A ⊂ X. The image of A under (or by) f , denoted by f [A], is the set of all
y ∈ Y such that y = f (x) for some x ∈ X; i.e.,
f [A] := {y ∈ Y : y = f (x), ∃ x ∈ X}.
• Let B ⊂ Y. The inverse image of B under (or by) f , denoted by f −1 [B], is the set
of all x ∈ X such that f (x) ∈ B; i.e.,
f −1 [B] := {x ∈ X : f (x) ∈ B}.
• A function f : X → Y is said to be onto (surjective) if f [x] = Y .
• A function f : X → Y is said to be one-to-one (injective) if f (x1 ) = f (x2 ) implies
x1 = x2 .
• Functions which are one-to-one from X onto Y are called one-to-one correspondences between X and Y (they are also called bijective).
• When f : X → Y is bijective, there exists a function g : Y → X (which is also
bijective) such that for all x ∈ X and for all y ∈ Y we have
g(f (x)) = x and
f (g(y)) = y.
In this case the function g is said to be the inverse of f and is denoted by
g = f −1 .
• If f : X → Y and g : Y → Z, we define new function h : X → Z by setting
h(x) = g(f (x)). h is called the composition of g with f and is denoted by g ◦ f.
• If f : X → Y and A ⊂ X, we define a function g : A → Y by g(x) = f (x)
whenever x ∈ A. The new function g : A → Y is called the restriction of f to A
and is denoted by f |A .
• A function f : X → Y is said to be
– increasing when x1 ≤ x2 =⇒ f (x1 ) ≤ f (x2 ).
– decreasing when x1 ≤ x2 =⇒ f (x1 ) ≥ f (x2 ).
Remark 1.5. One can treat n−tuples and sequences as functions. For instance, an
ordered pair can be considered as a function whose domain is the set {1, 2} and an n tuple
as a function whose domain is the set {1, 2, . . . , n}. Similarly, an infinite sequence is a
function with domain N.
Notation
• We denote n−tuples by (xi )ni=1 or simply (xi )n1 . Similarly, we usually denote a
∞
sequence by (xi )∞
i=1 or (xi )1 or (xi )i∈N .
• The range of a sequence is denoted by {xi }∞
i=1 or {xi }i∈N or even by {xi } when
there is no confusion.
Definition 1.6. A set is said to be
• countable if it is the range of some sequence.
• finite if it is a range of some finite sequence.
• uncountable if it is not countable.
• infinite if it is not finite.
4
Definition 1.7. Let f be an infinite sequence. We say that h is an infinite subsequence
of f if there is an increasing function g : N → N such that h = f ◦ g. If we write f as fi
and g as gi , then we denote h by fgi .
2. Measurable spaces
2.1. Preliminaries.
Definition 2.1. For Λ an index set. An indexed subset of X is a function on an index
set Λ to X. (If Λ = N, the notion of indexed sets coincides with notion of sequence.)
Definition 2.2. The union and intersection of an indexed set is defined as
[
^
Aλ = {x ∈ X : (∃λ)(λ ∈ Λ x ∈ Aλ )}
A∈Λ
\
Aλ = {x ∈ X : (∀λ)(λ ∈ Λ =⇒ x ∈ Aλ )}
λ∈Λ
Definition 2.3. For A, B ∈ P (X), if A ∩ B = ∅ we say A and B are disjoint sets. A
collection C of sets is said to be disjoint collection of sets or collection of pairwise disjoint
sets if any two sets in C are disjoint.
Theorem 2.4 (De’Morgan’s law). Let C be a collection of sets. Then
!c
!c
[
\
\
[
A =
Ac
A =
Ac
A∈C
A∈C
A∈C
A∈C
Definition 2.5. Let (An ) be a collection of subsets of X (or (An ) is a sequence in P (X)).
Then we define
∞ \
∞
∞ [
∞
[
\
Am .
Am ,
lim inf An :=
lim sup An :=
n→∞
n→∞
n=0 m=n
n=0 m=n
Theorem 2.6. If f maps X into Y and {Aλ }λ∈Λ is a collection of subsets of X, then
[
f [∪λ∈Λ Aλ ] =
f [Aλ ]
λ∈Λ
f [∩λ∈Λ Aλ ] ⊂
\
f [Aλ ]
λ∈Λ
If f maps X into Y and {Bλ }λ∈Λ is a collection of subsets of X, then
[
f −1 [∪λ∈Λ Bλ ] =
f −1 [Bλ ]
λ∈Λ
f
−1
[∩λ∈Λ Bλ ] =
\
λ∈Λ
f −1 [Bλ ]
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2.2. Algebra of Sets: Rings, algebras and σ−Algebras.
Definition 2.7. A non empty subset A of P (X) is said to be a ring if:
a.) ∅ ∈ A;
b.) A, B ∈ A =⇒ A ∪ B, A ∩ B ∈ A;
c.) A, B ∈ A =⇒ A − B ∈ A.
Definition 2.8. A collection G of subsets of X is called an algera of sets or Boolean
algebra if
a.) A ∪ B ∈ G whenever A, B ∈ G.
b.) Ac ∈ G whenever A ∈ G.
Exercise 2.9. Let G be an algebra of subsets of X. Show that G is a ring.
Theorem 2.10. Let G ⊂ P (X) be an algebra. Then
T
S
a.) For A1 , A2 , . . . , An ∈ G then we have ni=1 Ai , ni=1 Ai ∈ G.
b.) ∅, X ∈ G.
Proof. Proof will be done in class.
Theorem 2.11. Given any collection C of subsets of X, there is a a smallest algebra G
which contains C.
Proof. Will be done in class.
(Ai )ni=1
Theorem 2.12. Let G be an algebra of subsets and
Then there is a sequence (Bi )ni=1 of sets in G such that
Bn ∩ Bm = ∅
whenever
m 6= n and
be a sequence of sets in G.
∪ni=1 Bi = ∪ni=1 Ai
Proof. Will be proved in class.
Definition 2.13. An algebra G of sets is called a σ−algebra or a Borel field if every union
of a countable collection of sets in G is again in G; i.e., if (Ai )i∈N is a sequence of sets in
G then
[
Ai ∈ G.
i∈N
Exercise 2.14. Show that every countable intersection of sets in a σ−algebra G is again
in G.
Theorem 2.15. Given any collection C of subsets of X there is a smallest σ−algebra
which contains C; i.e., there is a σ−algebra, say G containing C such that if B is σ−algebra
containing C then G ⊂ B.
Proof. Will be proved in class.
Exercise 2.16. Let E be a σ−algebra and (An ) a sequence of sets in E. Show that both
lim sup An
n→∞
are in E.
and
lim inf An
n→∞
6
Exercise 2.17. Let X = R and let Σ = {A ⊂ R : A is countable or Ac is countable}.
Show that Σ is a σ−algebra.
Definition 2.18. Let C ⊂ P (X). Then the smallest σ-algebra containing C is called
a σ−algebra generated by C and it is denoted by σ(C). σ(C) is the intersection of all
σ−algebras containing C.
Definition 2.19. Let (X, d) be a metric space. The σ−algebra generated by all open
subsets of X is called the Borel σ−algebra of X and it is denoted by B(X).
Example 2.20. Let X = R. Let C be the set of all intervals of the form [a, b) with a ≤ b.
Then show that σ(C) = B(R).
2.3. Set Function.
Definition 2.21. A set function is a function whose domain is a collection of sets and
takes values in R.
Axiom 2.1 (Axiom of Choice). Let C be a collection of non-empty sets. Then there is
a function f defined on C whcih assigns to each set A ∈ C an element f (A) in A. The
function f is called a choice function.
Definition 2.22. Let G ⊂ P (X) be an σ−algebra and µ be a mapping from G to [0, ∞]
such that µ(∅) = 0. We say that the set function µ is
a.) additive if
A, B ∈ G, A ∩ B = ∅ =⇒ µ(A ∪ B) = µ(A) + µ(B).
b.) subadditive if
A, B ∈ G =⇒ µ(A ∪ B) ≤ µ(A) + µ(B).
Observe that if A ⊂ B and µ is additive, then we have B = A∪(B−A) and A∩(B−A) =
∅ and
(2.1)
µ(B) ≥ µ(A).
Therefore, one could conclude that an additive function is non-decreasing.
Exercise 2.23. Let G ⊂ P (X) be an algebra and {Ai }ni=1 ⊂ G be a pairwise disjoint
collection of sets in G. Show that
!
n
n
[
X
µ
Ai =
µ(Ai ).
i=1
i=1
(Hint: Use induction)
Definition 2.24. Let G ⊂ P (X) be an algebra and µ be a set function from G to [0, ∞]
such
that µ(∅) = 0. For any sequence (An ) ⊂ G of mutually disjoint sets such that
S
A
n n ∈ G,
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a.) The set function µ is called σ−additive (or countably
additive) if for any sequence
S
(An )n∈N ⊂ G of mutually disjoint sets such that n∈N An ∈ G we have
!
[
X
µ
An =
µ(An ).
n∈N
n∈N
b.) The set function µ is called σ−subadditive (or countably subadditive) if for any
sequence (An )n∈N ⊂ G we have
!
[
X
µ
An ≤
µ(An ).
n∈N
n∈N
Remark
2.25. Let G be an algebra, {An }n be a mutually disjoint collection in G such
S
that n An ∈ G and µ an additive set function on G. Then we have by (2.1) that
!
!
∞
k
∞
k
k
[
[
[
[
X
An ⊃
An =⇒ µ
An ≥ µ
An =
µ(An ), ∀ k ∈ N.
n=1
n=1
n=1
n=1
n=1
Now, letting the limit k → ∞ yields
!
∞
∞
[
X
µ
An ≥
µ(An ).
n=1
n=1
Thus to show, that an additive function is σ−additive, it is enough to prove that it is
σ−subadditive.
Exercise 2.26. Show that σ−additive set function are σ−subadditive.
Theorem 2.27 (Continuity on non-decreasing sequences). If µ is additive set function
on a σ−algebra G, then the following statements are equivalent.
1) µ is σ−additive.
2) (An ) ⊂ G and A ∈ G, An ↑ A =⇒ µ(An ) ↑ µ(A).
Proof. will be discussed in class.
Theorem 2.28 (Continuity over nonincreasing sequences). Let µ be a σ−additive set
function over a σ−algebra G Then
(An ) ⊂ G, A ∈ G, An ↓ A, µ(A0 ) < ∞ =⇒ µ(An ) ↓ µ(A).
Proof. will be discussed in class.
2.4. Measurable Spaces and Measure Spaces.
Definition 2.29. A measurable space is pair (X, Σ), where Σ is σ−algebra on X.
Definition 2.30. Let (X, Σ) be a measurable space. A measure on (X, Σ) is a function
µ : Σ → [0, ∞] such that
a.) µ(∅) = 0
8
b.) If (Ai ) is a sequence of mutually disjoint sets in Σ, then
∞
X
∞
µ(∪i=1 Ai ) =
µ(Ai )
i=1
Definition 2.31. A measure space is a triple (X, Σ, µ), where X is a set, Σ is a σ−algebra
on X and µ is a measure on Σ.
Definition 2.32. Let (X, Σ, µ) be a measure space. If µ(X) = 1, then (X, Σ, µ) is called
a probability space and µ is a probability measure.
Example 2.33. Let (X, Σ) be a measurable space.
a.) The set function ν : Σ → [0, ∞] defined by
m, if n(A) = m
ν(A) =
∞, otherwise
is a measure on X and is called a counting measure.
b.) Choose and fix x ∈ X. Define a set function µx : Σ → [0, ∞] by
1, if x ∈ A
µx (A) =
0, otherwise.
Then µx is a measure on X, called unit point mass measure.
Theorem 2.34. Let (X, Σ, µ) be a measure space. Then the following statements hold;
1) If A, b ∈ Σ such that A ⊂ B then µ(A) ≤ µ(B).
2) If (An )n is a sequence in Σ, then
∞
X
∞
µ(∪n=1 ) ≤
µ(An )
n=1
3) If (An ) is a sequence in Σ such that An ⊂ An+1 for all n, then
µ(∪∞
n=1 ) = lim µ(An )
n→∞
4) If (An ) is a sequence in Σ such that An ⊃ An+1 for all n, then
µ(∩∞
n=1 ) = lim µ(An )
n→∞
Proof. Will be discussed in class
Theorem 2.35 (Borel-Cantelli Lemma).
P∞ Let (X, Σ, µ) be a measure space. If (An ) is a
seqeucne of measurable sets such that n=1 µ(An ) < ∞, then µ(lim supn→∞ An ) = 0.
Proof. Will be discussed in class.
Definition 2.36. Let (X, Σ, µ) be a measure space. Then measure µ is said to be
1) finite if µ(X) < ∞.
2) σ−finite if there is a sequence (An ) of sets in Σ with µ(An ) < ∞ for all n such that
X = ∪∞
n=1 An .
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2.4.1. Ourter Measure.
Definition 2.37. Let X be a nonempty set. An outer measure on X is a set function
µ∗ : P (X) → [0, ∞] such that
a.) µ∗ (∅) = 0.
b.) If A ⊂ B, then µ∗ (A) ≤ µ∗ (B)
c.) For every sequence (An ) of subsets of X
!
∞
∞
[
X
An ≤
µ∗ (An )
µ∗
n=1
n=1
Example 2.38 (Examples of ourter measure).
a.) Let X 6= ∅. Define µ∗ : P (X) → [0, ∞] by
1, A 6= ∅
∗
µ (A) =
0, A = ∅
µ∗ is an outer measure.
b.) Let X be an uncountable set. Define µ∗ : P (X) → [0, ∞] by
0, A is countable 6= ∅
∗
µ (A) =
1, A otherwise
µ∗ is an outer measure.
Definition 2.39. Let X be a set and µ∗ be an outer measure on X. A subset E of X is
said to be µ∗ −measurable if for every subset A of X, we have
µ∗ (A) = µ∗ (A ∩ E) + µ∗ (A ∩ E c )
Remark 2.40. By definition of an outer measure, we have that
µ∗ (A) ≤ µ∗ (A ∩ E) + µ∗ (A ∩ E c ).
Therefore, to show that a subset E of X is µ∗ measurable, it suffices to show that
µ∗ (A) ≥ µ∗ (A ∩ E) + µ∗ (A ∩ E c )
Exercise 2.41. Prove that E is µ∗ −measurable if and only if E c is µ∗ −measurable.
Lemma 2.42. Let X be a set, µ∗ an outermeasure on X and E ⊂ X such that µ∗ (E) = 0.
Then E is µ∗ −measurable.
Proof. Will be discussed in class.
Lemma 2.43. Let X be a set and µ∗ be an outer measure on X. If E, F ⊂ X are µ∗
measurable, then E ∪ F is also µ∗ −measurable.
Proof. Will be discussed in class.
Exercise 2.44. Let X be a set and µ∗ be an outer measure on X. If E, F ⊂ X are µ∗
measurable, then E ∩ F is also µ∗ −measurable.
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Theorem 2.45. Let X be a set and µ∗ be an outer measure on X. Denote by M, the
collection of µ∗ −measurable subsets of X. Then M is an algebra.
Proof. will be discussed in class
Theorem 2.46. Let X be a set and µ∗ be an outermesure on X. Suppose that (En ) be a
sequence of mutually disjoint sets in M. Then for any A ⊂ X and any integer n ≥ 1, we
have
!
n
n
[
X
∗
µ A∩
Ej =
µ∗ (A ∩ Ej ).
j=1
j=1
Proof. will be discussed in class..
Exercise 2.47. Show that µ∗ is finitely addivitive on M.
Remark 2.48. Let X be a set and µ∗ be an outermeasure on X. Then the restriction
µ = µ∗ |M is a measure and is refereed to as a measure on X indeuced by the outermeasure
µ∗ .
2.4.2. Lebesgue Measure on R. Denote by L, the set of all open intervals in R. For
I = (a, b) ∈ L define `(I) = b − a =length of interval I.
Theorem 2.49. For any A ⊂ R, define m∗ : P (R) → [0, ∞] by
X
`(In ) : In ∈ L ∀n ∈ N&A ⊂ ∪∞
m∗ (A) = inf{
n=1 In }.
n
Then m∗ is an outer measure on R.
Proof.
a.) Since ∅ = (a, a) for all a ∈ R, we have by definition of m∗
0 ≤ m∗ (∅) ≤ `((a, a)) = 0 =⇒ m∗ (∅) = 0.
b.) Let A, B ⊂ R such that A ⊂ B. Let
β = {{In }n ⊂ L B ⊂ ∪n∈N In }
α = {{In }n ⊂ L| A ⊂ ∪n∈N In }
(
)
X
β̃ =
`(In )|{In }n ∈ β
n
α̃ =
(
X
)
`(In )|{In }n ∈ α
n
Clearly, α̃ ⊃ β̃ (Show!). Thus we have
inf β̃ ≥ inf α̃ =⇒ m∗ (B) ≥ m∗ (A).
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c.) Let {An } ⊂ R. If m∗ (An ) = ∞ for some n ∈ N, then nothing to prove. Suppose
m∗ (An ) < ∞ for all n. Then given > 0 there is a sequence (Ink )k of open intervals
in R such that An ⊂ ∪k Ink and
∞
X
`(Ink ) < m∗ (An ) + /2n
k=1
∗
=⇒ m (∪n An ) ≤
∞ X
∞ X
∞
X
`(Ink )
n=1 k=1 k=1
≤
∞
X
(m∗ (An ) + /2n )
n=1
=
∞
X
m∗ (An ) + .
n=1
Since > 0 arbitrary, we have
∗
m (∪n An ) ≤
∞
X
m∗ (An ).
n=1
∗
Thus from a), b) and c) one concludes m is an outer measure.
Definition 2.50. The outer measure m∗ defined in Theorem 2.49 is called Lebesgue
Outer Measure.
Definition 2.51. A set E ⊂ R is said to be Lebesgue measurable (measurable) is
m∗ (A) = m∗ (A ∩ E) + m∗ (A ∩ E c ).
Remark 2.52. Denote by M, the set of all Lebesgue measurable subset of R. Then then
M is a σ−Algebra and m = m∗ |M is a measure on M. The measure m is called a
Lebesgue measure.
2.5. Properties of Lebesgue Measure.
Proposition 2.53. The outer measure of an interval is its length. That is for I an
interval in R we have
m∗ (I) = `(I)
Proof. proved in class
Proposition 2.54. For any x ∈ R, m∗ ({x}) = 0.
Proof. Proved in class
Corollary 2.55. Every countable subset of R has outer measure zero.
Proposition 2.56. Lebesgue outer measure is transition invariant; i.e., m∗ (A + x) =
m∗ (A) for any x ∈ R and any set A ⊂ R.
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Proof. To be discussed in class
Proposition 2.57. For any set A ⊂ R and any > 0, there is an open set V such that
A ⊂ V and m∗ (V ) < m∗ (A) + .
Proof. To be discussed in class.
Proposition 2.58. For any set a ∈ R, the set (a, ∞) is measurable.
Proof. To be discussed in class
Exercise 2.59. Proof the following statements.
a.) Every interval in R is measurable.
b.) Every open set in R is measurable.
c.) Every closed set in R is measurable.
d.) Every set that is a countable union of closed sets in R is measurable.
e.) Every set that is a countable union of closed sets in R is measurable.
Definition 2.60.
a.) A set that is a union of countable collection of closed sets is
called Fσ − set (F −sigma).
b.) A set that is an intersection of a countable collection of open sets is called a Gδ −set
(G−delta).
Remark 2.61.
a.) Every Fσ -set is measurable.
b.) Every Gδ -set is measurable.
Proposition 2.62. Let E ⊂ R be m∗ lebesgue measurable on R. The following statements
are equivalent.
a.) E is measurable.
b.) For every > 0, there is an open set V such that E ⊂ V and m∗ (V \E) < .
c.) There is a Gδ -set G such that E ⊂ G and m∗ (G\E) = 0.
d.) For each , there is a closed set F such that F ⊂ E and m∗ (E\F ) < .
e.) There is an Fσ −set H such that H ⊂ E and m∗ (E\H) = 0
f.) (If m∗ (E) < ∞, then the above statements are equivalent to ) Given > 0, there
is a finite union U of open intervals such that m∗ (U ∆E) < .
Proof. To be discussed in class
3. The Lebesgue Integral
3.1. Measurable Functions.
Definition 3.1. Let (X, Σ) be a measurable space and E ∈ Σ. A function f : E →
[−∞, ∞] is said to be measurable, if for each α ∈ R the set {x ∈ E : f (x) > α} is
measurable.
Let (X, Σ) be a measurable space and E ∈ Σ and f : E → [−∞, ∞]. Then the following
statements are equivalent.
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a.)
b.)
c.)
d.)
13
f is measurable
For each α ∈ R, the set {x ∈ E : f (x) ≥ α} is measurable.
For each α ∈ R, the set {x ∈ E : f (x) < α} is measurable.
For each α ∈ R, the set {x ∈ E : f (x) ≤ α} is measurable.
Proposition 3.2.
Proof. to be discussed in class..
Example 3.3.
a.) The constant function is measurable
b.) Let (X, Σ) be a measurable space and let A ∈ Σ. The characterstic function on
A, χA is defined by
1, if x ∈ A
χA (x) =
0, otherwise
is measurable.
c.) Let B be a Borel σ−algebra in R and E ∈ B. Then any continuous function f :
E → [−∞, ∞] is measurable.
Proposition 3.4. Let (X, Σ) be a measurable space. If f and g are measurable real valued
functions defined on a common domain E ∈ Σ and c ∈ R. Then the following functions
are also measurable.
a.) f + c
b.) cf
c.) f ± g
d.) f 2
e.) f g
f.) |f |
g.) f ∨ g := max{f, g}
h.) f ∧ g := ∧{f, g}
Proof. to be discussed in class
Proposition 3.5. Let (X, Σ) be a measurable space. If (fn ) is a sequence of measurable
functions defined on a common domaing E ∈ Σ, then the functions
a.) supn fn
b.) inf n fn
c.) lim supn fn
d.) lim inf n fn
are all measurable.
Proof. to be discussed in class
Corollary 3.6. Let (X, Σ) be a measurable space. If f is a pointwise limit of sequence
(fn ) of measurable functions defined on a common domain E ∈ Σ, then f is measurable.
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Proof. Exercise..
Definition 3.7. Let f : X → (−∞, ∞) be a real valued function. The positive part f +
and negative part f − of f are defined by
f + = max{f, 0} = f ∨ 0
f − = max{−f, 0} = (−f ) ∨ 0
Exercise 3.8. Prove that f is measurable if and only if f + and f − are measurable.
Definition 3.9. Let (X, Σ) be a measurable space. A simple funcion on X is a function
of the form
n
X
φ=
cj χEj ,
j=1
where for each j = 1, 2, . . . cj ∈ [−∞, ∞] and Ej ∈ Σ.
Exercise 3.10. Prove that φ is measurable.
Theorem 3.11. Let (X, Σ) be a measurable space and f be a non-negative measurable
function. Then there is a monotonic inreasing sequence φn of non-negative simple functions which converge pointwise to f.
Proof. to be discussed in class..
Corollary 3.12. Let (X, Σ) be a measurable space and f be a measurable function. Then
there is a sequence of simple measurable fucntions which converge pointwise to f.
Proof. Exercise..
Definition 3.13. Let (X, Σ, µ) be a measure space and E ∈ Σ. A property p(x, ) where
x ∈ E is said to be almost everywhere (abbreviated, a.e.) on E if the set
N = {x ∈ E : p(x) fails } ∈ Σ
satisfies µ(N ) = 0.
Example 3.14.
a.) f = g a.e. if µ({x ∈ X : f (x) 6= g(x)}) = 0
b.) f ≤ g a.e. if µ({x ∈ X : f (x) > g(x)}) = 0
Definition 3.15. Let (X, Σ, µ) be a measure space. A function f : X → [−∞, ∞] is said
to be a.e. real valued, if µ({x ∈ X : |f (x)| = ∞}) = 0
A set of measure zero is called Null set.
Definition 3.16. A measure space (X, Σ, µ) is said to be complete if Σ contains all subsets
of X of measure zero. i.e., if E ⊂ X and µ(E) = 0 then E ∈ Σ.
Proposition 3.17. Let (X, Σ, µ) be a complete measure space and f = g a.e. If f is
measurable on E ∈ Σ, then so is g
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Proof. to be discussed in class..
Definition 3.18. Let (X, Σ, µ) be a measure space. A sequence (fn ) of almost everywhere
real valued functions on X is said to
a.) Converge almost everywhere to an a.e. real valued measurable function f ,
a.e.
denoted by fn −−→ f, if for each > 0 and each x ∈ X, there is a set E ∈ Σ and a
number N = N () ∈ N such that µ(E) < and
|fn (x) − f (x)| < ,
∀x ∈ X − E, ∀n ≥ N.
b.) Converge almost uniformly (a.u.) to an a.e. real valued measurable function
a.u.
f , denoted by fn −−→ f, if for each > 0 and each x ∈ X, there is a set E ∈ Σ and
a number N = N () ∈ N such that µ(E) < and
kfn − f kL∞ := sup |fn (x) − f (x)| < ,
∀n ≥ N.
x∈X−E
c.) Converge in measure to an a.e. real valued measurable function f , denoted by
µ
fn →
− f, if for each > 0
lim µ({x ∈ X : |fn (x) − f (x)| ≥ }) = 0
n→∞
(If µ is a probability measure, then this mode of convergence is called convergence
in probability)
Proposition 3.19. Let (X, Σ, µ) be a complete measure space and (fn ) be a sequence of
real valued measurable functions defined on E ∈ Σ which converges to f a.e. Then f is
measurable.
Proof. to be discussed in class.
Proposition 3.20. Let (X, Σ, µ) be a complete measure space and (fn ) be a sequence of
µ
a.u.
real valued measurable functions defined on X. If fn −−→ f , then fn →
− f
Proof. to be discussed in class..
Proposition 3.21. Let (X, Σ, µ) be a complete measure space and (fn ) be a sequence of
a.u.
a.e.
real valued measurable functions defined on X. If fn −−→ f , then fn −−→ f
Proof. to be discussed in class
Example 3.22. Let X = [0, ∞), µ =Lebesgue measure. For each n ∈ N, let fn = χn,n+ 1 .
n
Show that
a.e
a.) fn −→ 0
µ
b.) fn →
− 0
a.u
c.) fn −
6 →0
Two thorems with out proof.
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Theorem 3.23 (Egoroff’s Theorem). Let (X, Σ, µ) be a finite measure space and (fn ) be
a.e
a.u
a sequence of a.e. real valued measurale functions on X. If fn −→ f , then fn −→ f
Theorem 3.24 (Riesz Theorem). Let (X, Σ, µ) be a measure space and (fn ) be a sequence
µ
of a.e. real valued measurale functions on X. If fn →
− f , then there is a subsequence (fnk )
of (fn ) which converges almost everywhere to f.
3.2. Integrals.
3.2.1. Integral of non-negative simple functions.
Definition 3.25. Let φ be a non-negative simple function whith the canonical represenn
X
tation φ =
ai χEi . The Lebesgue integral of φ with respect to the measure µ, denoted
i=1
Z
by
φdµ, is the extended real number
X
Z
φdµ =
n
X
X
ai µ(Ei ).
i=1
If A ∈ Σ, we define
Z
Z
φdµ =
χA φdµ.
X
Z
φdµ < ∞.
The function φ is said to be integrable if
A
X
Z
Remark 3.26. A ∈ Σ =⇒
φdµ =
A
n
X
ai µ(A ∩ Ei ).
i=1
Proposition 3.27. et (X, Σ, µ) be a measure space, φ and ψ be non-negative simple
funcitons
positive
number. Then
Z and c a non-negative
Z
Z
(φ + ψ)dµ =
φdµ +
ψdµ
X
Z X
b.)
cφdµ = c
φdµ
X
XZ
Z
c.) if φ ≤ ψ, then
φdµ ≤
ψdµ
a.)
ZX
X
X
d.) IF A, B ∈ Σ such that A ∩ B = ∅, then
Z
Z
Z
ψdµ =
ψdµ +
ψdµ
A∪B
A
B
e.) The set function ν : Σ → [0, ∞] defined by
Z
ν(A) =
φdµ
A
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17
is a measure on X.
Z
f.) If A ∈ Σ and µ(A) = 0, then
φdµ = 0
A
Proof. to be discussed in class
Corollary 3.28. Let (X, Σ, µ) be a measure space. If A, B ∈ Σ such that A ⊂ B & µ(B −
A) = 0, then
Z
Z
φdµ =
φdµ
A
B
3.2.2. Integral of non-negative measurable functions.
Definition 3.29. Let (X, Σ, µ) be a measure space. Let f be a non-negative
measurable
R
function. The Lebesgue integral of f with respect to µ, denoted by X f dµ, is defined as
Z
Z
φdµ : 0 ≤ φ ≤ f, φ is a simple function .
f dµ = sup
X
X
If E ∈ Σ, then we define the Lebesgue integral of f over E with espect to µ as
Z
Z
χE f dµ.
f dµ =
X
E
R
The function f is said to be integrable (Lebesgue integrable) if X f dµ < ∞.
Proposition 3.30. Let f and g be non-negative measurable functions and c a non- negative real
Z number. Then
Z
a.)
cf dµ = c
f dµ
X
XZ
Z
gdµ
f dµ ≤
b.) If f ≤ g, then
X
X
Z
Z
c.) If A, B ∈ Σ such that A ⊂ B, then
f dµ ≤
f dµ.
A
Proof. to be discussed in class.
B