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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