Lecture-6 (Paper 1)

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Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
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
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Lebesgue Measurable Sets
Properties
Algebra of Sets
References
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Definition
A set E ⊆ R is said to be Lebesgue measurable if for any A ⊆ R
we have
\ \ m∗ (A) = m∗ A E + m∗ A E c .
(1)
We may write the above equality as a combination of
following two inequalities
1
2
T
T
m∗ (A) ≤ m∗ (A T E ) + m∗ (A T E c ).
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
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
E )c , we then have
h \ [ \ c i
m∗ (G ) = m∗ A E
A E
\ \ c
≤ m∗ A E + m ∗ A E .
Write, A = (A
T
E)
S
(A
T
(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 is
that 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
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
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,
T
T
A E ⊆ E ⇒ m∗ (A E ) ≤ m∗ (E ) and
T
T
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
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Suppose we are given with a measurable set, then what about its
complement?
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
\ \ c c
= m∗ A E c + m∗ A E c , ∵ E = E c .
Therefore, E c is measurable.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Continues.
Conversely, suppose that Ec is measurable. Then
\ \ c
m∗ (A) = m∗ A E c + m∗ A E c
\ \ c
c
= m∗ A E c + m∗ A E c
∵ Ec = E .
Hence, E is measurable.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Moreover, in the next theorem we see that union of two
measurable sets is again a measurable set.
Theorem
Let E1 and E2 be two measurable sets, then E1
Proof.
S
E2 is measurable.
Let A be any set of reals and E1 , E2 be two measurable sets. Since
E2 is m’able we have
\ \
\
\ \ m∗ A E1c = m∗ A E1c
E2 + m∗ A E1c
E2c .
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Coninues.
Now,
A
⇒
\
E1
[
E2
h \ i[h \ i
A E2
A E1
h \ i[h \ \ i
A E2 E1c .
= A E1
=
\ [ \ h \ \ i
m∗ A
E1 E2
≤ m∗ A E1 + m∗ A E2 E1c
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Continues.
Let us now consider
\ [ \ [ c m∗ A
+ m∗ A
E1 E2
E1 E2
\ \ ≤ m∗ A E1 + m∗ A E1c
= m∗ (A) .
Since E1 is also given measurable. Hence, E1
measurable.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
S
E2 is also
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Definition
A class a of sets is said to be an algebra if it satisfies the following
conditions:
1
If E ∈ a then E c ∈ a.
2
If E1 and E2 ∈ a, then E1
S
E2 ∈ a.
Thus a class a of sets is said to be algebra if it is closed under the
formation of complements or finite unions.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Lemma
Algebra is closed under the formation of finite intersections.
Proof.
Let A1 , A2 , .........., An ∈ a. Then,
\
[ [
[
\ c
Ac2
..... Acn .
A1 A2 ..... An = Ac1
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
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Continues.
Further, a being algebra is closed under the formation finite
unions. This implies that
[ [ [
Ac1
Ac2
... Acn ∈ a
∴
It follows that
n \ \ \ oc
A1 A2 ... An ∈ a.
A1
\
A2
\
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
...
\
An ∈ a.
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Definition (σ-Algebra)
A class a is said to be σ-algebra, if it is closed under the formation
of countable unions and of complements.
It is an easy exercise for the readers to verify that that
σ-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
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Theorem
Let A be any set of real number and let E1 , E2 , ..........., En be
pair-wise disjoint Lebesgue measurable sets then
!!
n
∞
\ X
\ [
∗
m ∗ A Ei .
m A
=
Ei
i=1
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
i=1
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
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
 

n
∞
\ X
[
∗



m An
m∗ An
En
Ei
=
j=1
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
j=1
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Continues.
Now, since Ei ’s are disjoint, we have


∞
\ [
\
\
A  En  En = A En .
j=1
And
A
\
"
Enc
n
\[
i=1
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
#
Ei == A
\
n−1
[
i=1
Ei
!
.
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Continues.
It follows that
m∗
A
" n
\ [
Ei
i=1
#
\
En
!
\ = m ∗ A En ,
and
m∗
A
" n
\ [
i=1
Ei
#
\
Enc
!
= m∗
A
\
En
"n−1
[
i=1
Ei
#!
.
Addition of above two equations leads us to the required
results.
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Theorem
Countable union of measurable sets is measurable.
Proof.
Let {A
Sn∞} be any countable condition of measurable sets and
E = n=1 An . We know that the class of Lebesgue measurable set
constitutes algebra. Therefore, there is a sequence {En } of
pair-wise disjoint measurable sets such that
E=
∞
[
An =
n=1
∞
[
En .
n=1
S∞
Let Fn = i=1 Ei , then Fn is measurable for each n and Fn ⊂ E .
This implies thatFnc ⊃ E c .
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Continues.
Moreover, if A be any set of real numbers then
\ \ A Fnc ⊃ A E c
⇒
\ \ m∗ A E c ≤ m∗ A Fnc .
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Continues.
Since, Fn is measurable we have
\ \ m∗ (A) ≥ m∗ A Fn + m∗ A Fnc ,
" n #!
\ \ [
∗
≥ m A
+ m∗ A E c ,
Ei
i=1
=
n
X
i=1
\ \ m ∗ A Ei + m ∗ A E c .
L.H.S. being independent of n, it follows that
m∗ (A) ≥
∞
X
i=1
\ \ m ∗ A Ei + m ∗ A E c
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
(3)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Continues.
Now,
A
"∞
\ [
i=1
Therefore
m∗
A
"∞
\ [
i=1
Ei
Ei
#!
#!
∞ \ [
A Ei .
=
i=1
= m∗
∞ \ [
A Ei
i=1
≤
∞
X
i=1
!
\ m ∗ A Ei
∞
\ \ X
⇒ m∗ A E ⇒
m ∗ A Ei
(4)
i=1
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Proof Continues.
Combining (3) and (4), it gives
\ \ m∗ (A) ≥ m∗ A E + m∗ A E c .
Hence,E =
S∞
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
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
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
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
G.de Barra : Measure theory and integration, New Age
International 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
Outline
Lebesgue Measurable Set
Properties
Algebra of Sets
Properties
References
Thank You !
Dr.A.Kaushik: Lecture-6 M.Sc.-I (Mathematics)
Directorate of Distance Education, K.U. Kurukshetra
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