Lebesgue Measurable Sets - Kurukshetra University · Lebesgue Measurable Set Properties Algebra of...

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




Lebesgue Measurable Sets

Properties

Algebra of Sets

References





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?





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)

.





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.





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.





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.





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

)

.





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

]





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.





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.





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.





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.





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.





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

)

.





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

)





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

)

.





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.





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 .





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

)

.





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)





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)





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.





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.





G.de Barra : Measure theory and integration, New AgeInternational Publishers.

A. Kaushik, Lecture Notes, Directorate of Distance Education,Kurukshetra University, Kurukshetra.





Thank You !


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