lebesgue measurable sets - kurukshetra university · lebesgue measurable set properties algebra of...
TRANSCRIPT
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Lebesgue Measurable Sets
Dr. Aditya Kaushik
Directorate of Distance Education
Kurukshetra University, Kurukshetra Haryana 136119 India
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Lebesgue Measurable Sets
Properties
Algebra of Sets
References
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Definition
A set E ⊆ R is said to be Lebesgue measurable if for any A ⊆ Rwe have
m∗ (A) = m∗
(
A⋂
E)
+ m∗
(
A⋂
E c)
. (1)
We may write the above equality as a combination offollowing two inequalities
1 m∗ (A) ≤ m∗ (A⋂
E ) + m∗ (A⋂
E c).2 m∗ (A) ≥ m∗ (A
⋂
E ) + m∗ (A⋂
E c).
Is there any wild guess !!! Why we are doing this?
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Write, A = (A⋂
E )⋃
(A⋂
E )c , we then have
m∗ (G ) = m∗
[(
A⋂
E)
⋃
(
A⋂
E)c]
≤ m∗
(
A⋂
E)
+ m ∗(
A⋂
E)c
. (2)
Are you able conclude anything from here? I think yes !!!If NO is your answer, I suggest you to look back to equality(1).Then, have a look at (2). What do you think is left to prove(1) if (2) holds eventually?
A necessary and sufficient condition for E to be measurable isthat for any set A ⊆ R
m∗ (A) ≥ m∗
(
A⋂
E)
+ m∗
(
A⋂
E c)
.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Let us begin with the very simple yet important lemma:
Lemma
If m∗ (E ) = 0, then E is measurable.
Proof.
Let A be any set of real numbers. Then,
A⋂
E ⊆ E ⇒ m∗ (A⋂
E ) ≤ m∗ (E ) and
A⋂
E c ⊆ A ⇒ m∗ (A⋂
E c) ≤ m∗ (A) .
Therefore,
m∗
(
A⋂
E)
+ m∗
(
A⋂
E c)
≤ m∗ (E ) + m∗ (A) ,
= 0 + m∗ (A) .
Hence, E is measurable.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Suppose we are given with a measurable set, then what about itscomplement?
Lemma
E is measurable iff Ec is measurable.
Proof.
Let A ⊆ R and E be a measurable set. Then,
m∗ (A) = m∗
(
A⋂
E)
+ m∗
(
A⋂
E c)
= m∗
(
A⋂
E c)
+ m∗
(
A⋂
E cc)
,
[
∵ E = E cc ]
.
Therefore, E c is measurable.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Continues.
Conversely, suppose that Ec is measurable. Then
m∗ (A) = m∗
(
A⋂
E c)
+ m∗
(
A⋂
E cc)
= m∗
(
A⋂
E c)
+ m∗
(
A⋂
E cc)
[
∵ E cc
= E]
.
Hence, E is measurable.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Moreover, in the next theorem we see that union of twomeasurable sets is again a measurable set.
Theorem
Let E1 and E2 be two measurable sets, then E1
⋃
E2 is measurable.
Proof.
Let A be any set of reals and E1, E2 be two measurable sets. SinceE2 is m’able we have
m∗
(
A⋂
E c1
)
= m∗
(
A⋂
E c1
⋂
E2
)
+ m∗
(
A⋂
E c1
⋂
E c2
)
.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Coninues.
Now,
A⋂
(
E1
⋃
E2
)
=[
A⋂
E1
]
⋃
[
A⋂
E2
]
=[
A⋂
E1
]
⋃
[
A⋂
E2
⋂
E c1
]
.
⇒ m∗
(
A⋂
(
E1
⋃
E2
))
≤ m∗
(
A⋂
E1
)
+ m∗
[
A⋂
E2
⋂
E c1
]
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Continues.
Let us now consider
m∗
(
A⋂
(
E1
⋃
E2
))
+ m∗
(
A⋂
(
E1
⋃
E2
)c)
≤ m∗
(
A⋂
E1
)
+ m∗
(
A⋂
E c1
)
= m∗ (A) .
Since E1 is also given measurable. Hence, E1
⋃
E2 is alsomeasurable.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Definition
A class a of sets is said to be an algebra if it satisfies the followingconditions:
1 If E ∈ a then E c ∈ a.
2 If E1 and E2 ∈ a, then E1
⋃
E2 ∈ a.
Thus a class a of sets is said to be algebra if it is closed under theformation of complements or finite unions.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Lemma
Algebra is closed under the formation of finite intersections.
Proof.
Let A1, A2, ..........,An ∈ a. Then,
(
A1
⋂
A2.....
⋂
An
)c
= Ac1
⋃
Ac2
⋃
.....
⋃
Acn.
Now, An ∈ a ∀n, then Acn ∈ a because a is algebra and is therefore
closed under the formation of compliments.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Continues.
Further, a being algebra is closed under the formation finiteunions. This implies that
Ac1
⋃
Ac2
⋃
...
⋃
Acn ∈ a
∴
{
A1
⋂
A2
⋂
...
⋂
An
}c
∈ a.
It follows thatA1
⋂
A2
⋂
...
⋂
An ∈ a.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Definition (σ-Algebra)
A class a is said to be σ-algebra, if it is closed under the formationof countable unions and of complements.
It is an easy exercise for the readers to verify that thatσ-algebra is closed under the formation of finite intersection.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Theorem
Let A be any set of real number and let E1, E2, ...........,En bepair-wise disjoint Lebesgue measurable sets then
m∗
(
A⋂
(
∞⋃
i=1
Ei
))
=n∑
i=1
m∗
(
A⋂
Ei
)
.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof.
We prove the result using mathematical induction on n.
For, n = 1
m∗
(
A⋂
E1
)
= m∗
(
A⋂
E1
)
Thus, the result is true for n = 1.
Suppose that the result is true for (n-1) sets Ei then we have
m∗
An
∞⋃
j=1
En
=n∑
j=1
m∗
(
An
⋂
Ei
)
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Continues.
Now, since Ei ’s are disjoint, we have
A⋂
∞⋃
j=1
En
⋂
En = A⋂
En.
And
A⋂
[
E cn
⋂
n⋃
i=1
Ei
]
== A⋂
(
n−1⋃
i=1
Ei
)
.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Continues.
It follows that
m∗
(
A⋂
[
n⋃
i=1
Ei
]
⋂
En
)
= m∗
(
A⋂
En
)
,
and
m∗
(
A⋂
[
n⋃
i=1
Ei
]
⋂
E cn
)
= m∗
(
A⋂
En
[
n−1⋃
i=1
Ei
])
.
Addition of above two equations leads us to the requiredresults.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Theorem
Countable union of measurable sets is measurable.
Proof.
Let {An} be any countable condition of measurable sets andE =
⋃
∞
n=1 An. We know that the class of Lebesgue measurable setconstitutes algebra. Therefore, there is a sequence {En} ofpair-wise disjoint measurable sets such that
E =∞⋃
n=1
An =∞⋃
n=1
En.
Let Fn =⋃
∞
i=1 Ei , then Fn is measurable for each n and Fn ⊂ E .This implies thatF c
n ⊃ E c .
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Continues.
Moreover, if A be any set of real numbers then
(
A⋂
F cn
)
⊃(
A⋂
E c)
⇒ m∗
(
A⋂
E c)
≤ m∗
(
A⋂
F cn
)
.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Continues.
Since, Fn is measurable we have
m∗ (A) ≥ m∗
(
A⋂
Fn
)
+ m∗
(
A⋂
F cn
)
,
≥ m∗
(
A⋂
[
n⋃
i=1
Ei
])
+ m∗
(
A⋂
E c)
,
=n∑
i=1
m∗
(
A⋂
Ei
)
+ m∗
(
A⋂
E c)
.
L.H.S. being independent of n, it follows that
m∗ (A) ≥∞∑
i=1
m∗
(
A⋂
Ei
)
+ m∗
(
A⋂
E c)
(3)
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Continues.
Now,(
A⋂
[
∞⋃
i=1
Ei
])
=∞⋃
i=1
(
A⋂
Ei
)
.
Therefore
m∗
(
A⋂
[
∞⋃
i=1
Ei
])
= m∗
(
∞⋃
i=1
(
A⋂
Ei
)
)
≤∞∑
i=1
m∗
(
A⋂
Ei
)
⇒ m∗
(
A⋂
E)
⇒
∞∑
i=1
m∗
(
A⋂
Ei
)
(4)
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Proof Continues.
Combining (3) and (4), it gives
m∗ (A) ≥ m∗
(
A⋂
E)
+ m∗
(
A⋂
E c)
.
Hence,E =⋃
∞
i=1 Ei is measurable.
As a consequence of result we just proved, we have
Corollary
The class of Lebesgue measurable sets is a σ algebra.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Let us end this lecture with the statement of an important results.
Theorem
Interval (a,∞) is measurable.
Proof of the above theorem is left for the readers as an exercise.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
G.de Barra : Measure theory and integration, New AgeInternational Publishers.
A. Kaushik, Lecture Notes, Directorate of Distance Education,Kurukshetra University, Kurukshetra.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra
OutlineLebesgue Measurable Set
PropertiesAlgebra of Sets
PropertiesReferences
Thank You !
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics) Directorate of Distance Education, K.U. Kurukshetra