Microsoft Word - 72-80   72  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020       𝛌– 𝐀𝐥𝐠𝐞𝐛𝐫𝐚 with Some of Their Properties Hassan Hussien Ebrahim Rusul Abd Al_salam Ali Article history: Received 23 June 2019, Accepted 19 August 2019, Published in April 2020. Abstract The objective of this paper is, firstly, we study a new concept noted by λ–algebra and discuss the properties of this concept. Secondly, we introduce a new concept related to the λ–algebra such as smallest λ–algebra. Thirdly, we introduce the notion of the restriction of λ–algebra on a nonempty subset 𝔇 of 𝔓 and investigate some of its basic properties. Furthermore, we present the relationships between α– σ–field, monotone class, β– σ–field and λ–algebra. Finally, we introduce the concept of measure relative to the λ–algebra and prove that every measure relative to the λ– algebra is complete. Keywords: σ–field, increasing sequence, α– σ–field, monotone class, β– σ–field. 1. Introduction About forty seven year ago, Robert [1]. Studied the concept of σ–field, where a collection 𝒦 is called σ–field of a set 𝔓 if 𝔓ϵ𝒦 and 𝒦 is closed under complementation and countable union. Many authors studied the concept of σ–field, for example see [2-4]. And [5]. The notion of increasing sequence and decreasing sequence studied by Robert, where D , D , … are subsets of a set 𝔓, if D ⊂ D ⊂ ⋯ and ⋃ D D. Then we say that D increase toD; we write D ↑ D. If D ⊃ D ⊃ ⋯ and ⋂ D D , we say that D decrease toD; we write D ↓ D [1]. Zhenyuan and George in 2009 studied the concept of monotone class which represents the generalization of σ–field, where a collection 𝒦 of subsets of a nonempty set 𝔓 is said to be monotone class iff whenever D , D , … ϵ𝒦 such that D ↑ D, then Dϵ 𝒦 and if D ↓ D, then Dϵ 𝒦 [6]. In 2019, Ibrahim and Hassan introduced some concepts such as α– σ–field and β– σ–field which represent the generalizations of σ–field, where a collection 𝒦 is said to be α– σ–field iff Φ, 𝔓ϵ 𝒦 and 𝒦 is closed under countable union [7]. And a collection 𝒦 is said to be β– σ–field if Φ, 𝔓ϵ 𝒦 and 𝒦 is closed under countable intersection [7]. Ibrahim and Hassan in 2019 also introduced the concept of δ–field as a stronger form of these concepts, where a collection 𝒦 is said to δ–field iff Φϵ 𝒦 and if Φ Aϵ𝒦 andA ⊂ B ⊆ 𝔓, then Bϵ𝒦and 𝒦 is closed under countable intersection [8]. The concept of complete measure on Department of Mathematic /College of Computer science and Mathematics /Tikrit University/ Tikrit/Iraq. hassan1962pl@tu.edu.iq  Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.2.2428 68@gmail.comrusulsalam6   73 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 σ–field was studied by Robert in 1972, but not necessarily that every measure defined on σ–field is complete. In this work, we prove that every measure defined on λ– algebra is complete. The main aim of this paper is to introduce and study new concept such as λ– algebra as a stronger from of α– σ–field and monotone class. And we give basic properties and examples of this concept. 2. The main results: Let P(𝔓) denoted to the power set of a nonempty set 𝔓 and we start this section by the definition of λ– algebra. Definition 1 A nonempty collection 𝒦 of a set 𝔓, 𝒦 𝔓 is called λ– algebra or (λ– field of a set 𝔓 if: 1- 𝔓ϵ𝒦. 2- If Dϵ𝒦 and E ⊂ D ⊂ 𝔓, then Eϵ𝒦. 3- If D , D , … ϵ𝒦, then ⋃ D ϵ𝒦. Definition 2 If 𝒦 is a λ– algebra of a set 𝔓 .Then a pair (𝔓 , 𝒦) is called measurable space relative to the λ– algebra 𝒦 and the elements of 𝒦 are called the measurable sets. Example 3 Let 𝔓 ={1,2,3,4} and 𝒦 { Φ,{1},{2},{4},{1,2},{1,4},{2,4},{1,2,4},𝔓 }. Then (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦. Proposition 4 For any λ– algebra 𝒦 of a set𝔓, the following hold: 1- Φϵ𝒦 2- If D , D , … , D ϵ𝒦, then ⋃ D ϵ𝒦. 3- If D , D , … ϵ𝒦, then ⋂ D ϵ𝒦. 4- If D , D , … , D ϵ𝒦, then ⋂ D ϵ𝒦. Proof The proof follows from definition of λ– algebra. Lemma 5 Let 𝒦 ∈ be a collection of λ– algebra on 𝔓. Then ⋂ 𝒦∈ is a λ– algebra on 𝔓. Proof Since 𝒦 is λ– algebra ∀ α ∈ Ι, then 𝔓ϵ𝒦 ∀ α ∈ Ι, hence 𝒦 Φ ∀α ∈ Ι and ⋂ 𝒦∈ Φ, therefore 𝔓 ϵ ⋂ 𝒦∈ . Let Dϵ ⋂ 𝒦∈ and E ⊂ D ⊂ 𝔓 , then Dϵ𝒦 ∀α ∈ Ι, but 𝒦 is λ– algebra ∀ α ∈ Ι and E ⊂ D. So, we get Eϵ𝒦 ∀α ∈ Ι, hence Eϵ ⋂ 𝒦∈ . Let D , D , … ϵ ⋂ 𝒦∈ .Then, D , D , … ϵ𝒦 , ∀α ∈ Ι, but 𝒦 is λ– algebra ∀ α ∈ Ι which implies that ⋃ D ϵ𝒦 , ∀α ∈ Ι, hence ⋃ D ϵ ⋂ 𝒦∈ . Therefore, ⋂ 𝒦∈ is a λ– algebra.   74 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Definition 6 Let 𝒥 ⊆ P 𝔓 . Then the intersection of all λ– algebra of 𝔓 which includes 𝒥 is called the λ– algebra generated by 𝒥 and denoted by λ 𝒥 , that is, λ 𝒥 = ⋂ 𝒦 : 𝒦 is a λ – algebra of 𝔓 and ⊆ 𝒦 , ∀α ∈ Ι . Proposition 7 Let 𝒥 ⊆ P 𝔓 . Then λ 𝒥 is the smallest λ– algebra of 𝔓 which includes 𝒥. Proof Since λ 𝒥 =⋂ 𝒦 : 𝒦 is a λ – algebra of 𝔓 and 𝒥 ⊆ 𝒦 , ∀α ∈ Ι . Then λ 𝒥 is λ– algebra of 𝔓 by Lemma 5. To prove λ 𝒥 ⊇ 𝒥, let each of 𝒦 is a λ– algebra of 𝔓 and 𝒥 ⊆ 𝒦 , ∀α ∈ Ι. Then 𝒥 ⊆ ⋂ 𝒦∈ , therefore 𝒥 ⊆ λ 𝒥 . Now, let 𝒦 ∗ is a λ – algebra of 𝔓 such that 𝒦 ∗ ⊇ 𝒥. Then ⋂ 𝒦 : 𝒦 is a λ – algebra of 𝔓 and 𝒥 ⊆ 𝒦 , ∀α ∈ Ι ⊆ 𝒦 ∗, hence λ 𝒥 ⊆ 𝒦 ∗. Therefore, λ 𝒥 is the smallest λ– algebra of 𝔓 which includes 𝒥. If we take Example 3 and if we assume 𝒥 ={{1},{2}}, then λ 𝒥 ={Φ,{1},{2},{1,2}, 𝔓} is the smallest λ– algebra of a set 𝔓 which includes 𝒥. Theorem 8 Let 𝒥 ⊆ P 𝔓 . Then (𝔓 , 𝒥) is measurable space relative to the λ– algebra 𝒥. if and only if 𝒥 λ 𝒥 . Proof Suppose that (𝔓 , 𝒥) is (a) measurable space relative to the λ– algebra 𝒥. From Proposition 7, we have λ 𝒥 is the smallest λ –algebra of a set 𝔓 which includes 𝒥 implies that𝒥 ⊆ λ 𝒥 . By hypothesis, we have 𝒥 is a λ– algebra of a set𝔓, but 𝒥 ⊆ 𝒥 and λ 𝒥 is the smallest λ –algebra of a set 𝔓 which includes 𝒥, then λ 𝒥 ⊆ 𝒥 , hence 𝒥 λ 𝒥 . Conversely) Let 𝒥 ⊆ P 𝔓 and let 𝒥 λ 𝒥 . Since λ 𝒥 is a λ– algebra of a set 𝔓, then 𝒥 is λ– algebra of a set 𝔓. If we take Example 3 and if we assume 𝒥 ={Φ,{1},𝔓}, then we conclude that λ 𝒥 = 𝒥. Now, we introduce the notion of restriction and study the basic properties of this notion. Definition 9 Let 𝒦 ⊆ P 𝔓 and Φ 𝔇 ⊆ 𝔓 . Then, the restriction of 𝒦 over the set 𝔇 is denoted by 𝒦|𝔇 and defined as follows: 𝒦|𝔇 = {B: B=E⋂𝔇, for some Eϵ 𝒦}. Proposition10 Let (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦 and Φ 𝔇 ⊆ 𝔓. Then 𝒦|𝔇 = {E ⊆ 𝔇: Eϵ 𝒦}. Proof Let Bϵ 𝒦|𝔇 . Then B=E⋂𝔇, for some Eϵ𝒦. Since E⋂𝔇 ⊆ E and 𝒦 is λ –algebra of a set 𝔓, then E⋂𝔇ϵ𝒦, hence Bϵ 𝒦. Since, E⋂𝔇 ⊆ 𝔇, then B⊆ 𝔇. Therefore Bϵ{𝐸 ⊆ 𝔇:Eϵ 𝒦} and 𝒦|𝔇 ⊆{A ⊆ 𝔇:Aϵ 𝒦}. Let Cϵ{𝐸 ⊆ 𝔇 : Eϵ 𝒦}. Then, C ⊆ 𝔇, and C ϵ 𝒦, hence,   75 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 C=C⋂ 𝔇, but Cϵ 𝒦, then Cϵ 𝒦|𝔇 which implies that{𝐸 ⊆ 𝔇:Eϵ 𝒦}⊆ 𝒦|𝔇 , therefore 𝒦|𝔇 ={A ⊆ 𝔇:Aϵ 𝒦}. Corollary 11 Let (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦 and Φ 𝔇 ⊆ 𝔓. Then 𝒦|𝔇 ⊆ 𝒦. Proof The result follows from Proposition10 Proposition 12 Let (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦,and 𝔇 ⊆ 𝔓. Then (𝔇 , 𝒦|𝔇) is measurable space relative to the λ– algebra 𝒦𝔇 Proof Since (𝔓 , 𝒦) is measurable space relative to theλ– algebra 𝒦 , then 𝔓 ϵ 𝒦. Since ⊆ 𝔓 , then 𝔇 𝔓 ⋂𝔇 and 𝔇 ϵ 𝒦|𝔇. Let Bϵ 𝒦|𝔇and F ⊂ B ⊂ 𝔇. Then by Corollary 11, we get Bϵ 𝒦. But F ⊂ B ⊂ 𝔇 ⊂ 𝔓 and (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦, then Fϵ 𝒦. Now, F ⊂ 𝔇, and Fϵ 𝒦, then by Proposition 10, we have Fϵ 𝒦|𝔇. Let B , B , … ϵ 𝒦|𝔇. Then there exist E , E , … ϵ𝒦 such that B =E ⋂ 𝔇 where i=1,2,…, hence ⋃ B =⋃ E ⋂ 𝔇 = ⋃ E ⋂ 𝔇. But (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦 and E , E , … ϵ𝒦, then, ⋃ E ϵ𝒦. Hence, ⋃ B ϵ 𝒦|𝔇. Therefore, (𝔇 , 𝒦|𝔇) is measurable space relative to the λ– algebra 𝒦|𝔇. Example 13 Let 𝔓 ={1,2,3,4,5} and 𝒦 { Φ,{1},{3},{5},{1,3},{1,5},{3,5},{1,3,5},𝔓 }. Then (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦. If 𝔇 {1,2,4}, then 𝒦|𝔇={Φ,{1},𝔇}, hence (𝔇 , 𝒦|𝔇) is measurable space relative to the λ– algebra 𝒦|𝔇 and 𝒦|𝔇 ⊆ 𝒦. Proposition 14 Let 𝒥 ⊆ P 𝔓 and Φ 𝔇 ⊆ 𝔓. If 𝒦 is a λ –algebra of 𝔓 which includes 𝒥, then λ 𝒥 |𝔇 is a λ– algebra of a set 𝔇. Proof The result follows from Proposition 7 and Proposition 12. Proposition 15 Let 𝒥 ⊆ P 𝔓 and Φ 𝔇 ⊆ 𝔓 and 𝒥|𝔇 is the restriction of 𝒥 over the set 𝔇. Then λ 𝒥|𝔇 is the smallest λ– algebra of a set𝔇, which includes 𝒥|𝔇, where λ 𝒥|𝔇 = ⋂ 𝒦 |𝔇: 𝒦 |𝔇 is a λ -algebra of 𝔇 , and 𝒦 |𝔇 ⊇ 𝒥|𝔇, ∀i ∈ Ι}. Proof From Lemma 5, we get λ 𝒥|𝔇 is a λ– algebra of a set𝔇. To prove that λ 𝒥|𝔇 ⊇ 𝒥|𝔇, suppose that each of 𝒦 |𝔇 is a λ -algebra of a set 𝔇 and𝒦 |𝔇 ⊇ 𝒥|𝔇, ∀i ∈ Ι, then 𝒥|𝔇 ⊆ ⋂ 𝒦 |𝔇 ∈ , hence 𝒥|𝔇 ⊆ λ 𝒥|𝔇 . Now, let 𝒦 ∗|𝔇 is a λ– algebra of a set 𝔇 such that 𝒦 ∗|𝔇 ⊇ 𝒥|𝔇. Then 𝒦 ∗|𝔇 ⊇ λ 𝒥|𝔇 . Therefore, λ 𝒥|𝔇 is the smallest λ– algebra of a set 𝔇 includes𝒥|𝔇.   76 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Proposition 16 Let 𝒥 ⊆ P 𝔓 and Φ 𝔇 ⊆ 𝔓, define the collection 𝒦 as: 𝒦 ={E ⊆ 𝔓: (E⋂ 𝔇 ϵ λ 𝒥|𝔇 }. Then (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦. Proof Since λ 𝒥|𝔇 } is a λ– algebra of a set 𝔇, then Φ, 𝔇 ϵ λ 𝒥|𝔇 . Since 𝔇 ⊆ 𝔓, then 𝔇 = 𝔓 ⋂ 𝔇 and 𝔓 ϵ𝒦. Let Eϵ𝒦 and F ⊂ E ⊂ 𝔓. Then, E⋂ 𝔇 ϵλ 𝒥|𝔇 . Since, F ⊂ E, then F⋂ 𝔇 ⊂ E⋂ 𝔇 . But λ 𝒥|𝔇 is a λ– algebra of a set𝔇, which implies that F⋂𝔇 ϵ λ 𝒥|𝔇 and Fϵ𝒦. Let E , E , … ϵ 𝒦. Then E ⋂𝔇 ϵ λ 𝒥|𝔇 , for all i=1,2,…, hence ⋃ E ⋂ 𝔇 ϵλ 𝒥|𝔇 and ⋃ E ⋂ 𝔇 ϵλ 𝒥|𝔇 implies that ⋃ E ϵ 𝒦. Therefore 𝒦 is λ–algebra of a set 𝔓. Theorem 17 Let 𝒥 ⊆ P 𝔓 and Φ 𝔇 ⊆ 𝔓. Then λ 𝒥|𝔇 =λ 𝒥 |𝔇. Proof Let Bϵ𝒥|𝔇, then B=E⋂ 𝔇, for someEϵ 𝒥. But 𝒥 ⊆ λ 𝒥 , then Eϵ λ 𝒥 , thus Bϵ λ 𝒥 |𝔇, hence 𝒥|𝔇 ⊆ λ 𝒥 |𝔇, but λ 𝒥|𝔇 is smallest λ –algebra of a set 𝔇, which include 𝒥|𝔇 and λ 𝒥 |𝔇is a λ– algebra of a set 𝔇 which include 𝒥|𝔇, then λ 𝒥|𝔇 ⊆ λ 𝒥 |𝔇. Now, define collection 𝒦 as: 𝒦= {E ⊆ 𝔓 : E⋂ 𝔇 ϵλ 𝒥|𝔇 }, then from Proposition 16, we obtain 𝒦 is a λ– algebra of a set 𝔓. Let Cϵ 𝒥, then C ∩ 𝔇 ϵ𝒥|𝔇, but 𝒥|𝔇 ⊆ λ 𝒥|𝔇 implies that C ∩ 𝔇 ϵ λ 𝒥|𝔇 , hence Cϵ 𝒦 and 𝒥 ⊆ 𝒦. Let Bϵ λ 𝒥 |𝔇, then B= F ∩ 𝔇, for some Fϵ λ 𝒥 . But λ 𝒥 ⊆ 𝒦, then Fϵ 𝒦, hence Bϵ λ 𝒥|𝔇 and λ 𝒥 |𝔇 ⊆ λ 𝒥|𝔇 , consequently λ 𝒥|𝔇 = λ 𝒥 |𝔇. We end this section by introduce the relationships between α– σ–field, monotone class, β– σ–field and λ– algebra. Proposition 18 Every λ– algebra is a α– σ–field. Proof Let 𝒦 be a λ– algebra of a set 𝔓. Then by definition of λ– algebra, we have Φ, 𝔓ϵ𝒦. Let D , D , … ϵ𝒦. Since 𝒦 is a λ– algebra, then by definition of 𝒦, we have ⋃ D ϵ 𝒦. Therefore 𝒦 is a α– σ–field. In general, the converse of above proposition is not true. For example, if 𝔓 ={1,2,3} and 𝒦 { Φ ,{1},{1,3},𝔓 }, then 𝒦 is α– σ– field but not λ– algebra, because {1,3}∈ 𝒦 and {3}⊂{1,3}, but {3}∉ 𝒦. Proposition 19 Every λ– algebra is a β– σ–field. Proof The proof follows from Proposition 4 and definition of λ– algebra. In general, the converse of above proposition is not true as shown in following example.   77 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Example 20 Let 𝔓 ={1,2,3,4} and 𝒦 { Φ ,{1},{1,3,4},{3,4},𝔓 }. Then, 𝒦 is β– σ– field but not λ– algebra, because {1,3,4}∈ 𝒦 and {3,4}⊂{1,3,4}, but {3,4}∉ 𝒦. Proposition 21 Every λ– algebra is a monotone class. Proof Let 𝒦 be a λ– algebra of a set 𝔓 and D , D , … ϵ𝒦 such thatD ↑ D. Then ⋃ D D Since 𝒦 is a λ– algebra, then by definition of 𝒦, we have ⋃ D ϵ 𝒦 which implies that Dϵ 𝒦. Let D , D , … ϵ𝒦 such that D ↓ D. Then, ⋂ D D, but 𝒦 is a λ– algebra, implies that ⋂ D ϵ𝒦 and D ϵ𝒦. Hence 𝒦 is a monotone class. In general, the converse of above proposition is not true. For example, if 𝔓 ={1,2,3} and 𝕄 { Φ,{1},{1,2} }, then 𝕄 is a monotone class, but not λ– algebra, because {1,2}∈ 𝕄 and {2}⊂{1,2}, but {2}∉ 𝕄. Definition 22 [6] Let 𝒥 ⊆ P 𝔓 . Then the intersection of all monotone classes of 𝔓 which include 𝒥 is called the monotone class generated by 𝒥 and denoted by 𝕄 𝒥 , that is, 𝕄 𝒥 = ⋂ 𝕄 : 𝕄 is a monotone class of 𝔓 and 𝒥 ⊆ 𝕄 , ∀i ∈ Ι . Lemma 23 [6] Let 𝕄 ∈ be a collection of monotone classes on 𝔓. Then ⋂ 𝕄∈ is a monotone class on 𝔓. Proposition 24 [6] Let 𝒥 ⊆ P 𝔓 . Then 𝕄 𝒥 is the smallest monotone class of 𝔓 which includes 𝒥. Theorem 25 Let 𝒥 ⊆ P 𝔓 . Then 𝕄 𝒥 ⊆ λ 𝒥 . Proof Let 𝒥 ⊆ P 𝔓 . Then by Proposition 7, we have λ 𝒥 is a λ– algebra of 𝔓 which includes 𝒥. From Proposition 21, we have, every λ– algebra is a monotone class, implies that λ 𝒥 is a monotone class which includes 𝒥. But 𝕄 𝒥 is the smallest monotone class which includes 𝒥 by Proposition 24, then 𝕄 𝒥 ⊆ λ 𝒥 . 3. Measure Defined on 𝛌– 𝐚𝐥𝐠𝐞𝐛𝐫𝐚 Our aim in this section is to prove that any measure defined on λ– algebra is complete. We begin with the notions of measure on λ– algebra. Definition 26 Let (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦. Then, a set function 𝔐, 𝔐: 𝒦 → 0 , ∞ is called measure relative to the λ– algebra 𝒦 if whenever 𝐷 , 𝐷 , … form a finite or countably infinite collection of disjoint sets in 𝒦, we have 𝔐 ⋃ 𝐷 ∑ 𝔐 𝐷 and 𝔐 Φ 0.   78 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Example 27 Let 𝔓 ={1,2,3} and 𝒦 { Φ,{1},{3},{1,3},𝔓 }. Then (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦. If we define a set function 𝔐: 𝒦 → 0 , ∞ by 𝔐(𝐷) = 𝑜 ; 𝑖𝑓 𝐷 Φ ; 𝑖𝑓 𝐷 1 𝑜𝑟 3 1 ; 𝑜𝑡ℎ𝑒𝑟 𝑤𝑖𝑠𝑒 Then 𝔐 is a measure relative to the λ– algebra 𝒦. Definition 28 A measure space relative to the λ– algebra 𝒦 is a triple (𝔓 , 𝒦, 𝔐) where (𝔓 , 𝒦) is measurable space relative to the λ– algebra 𝒦 and 𝔐 is a measure relative to the λ– algebra 𝒦. In the following Theorem, we use mathematical induction to prove that the linear combination of measure relative to the λ– algebra 𝒦 is also measure relative to the λ– algebra 𝒦. Theorem 29 Let (𝔓 , 𝒦, 𝔐 ) be a measure space relative to the λ– algebra 𝒦 and 𝑐 ∈ 0, ∞ for all 𝑗 1,2, … , 𝑘. If a set function ∑ 𝑐 𝔐 : ℘ → 0, ∞ is defined by: ∑ 𝑐 𝔐 𝐷 ∑ 𝑐 . 𝔐 𝐷 ∀𝐷𝜖℘, then (𝔓 , 𝒦, ∑ 𝑐 𝔐 ) is measure space relative to the λ– algebra 𝒦. Proof If 𝑘 2, then 𝑐 𝔐 𝑐 𝔐 Φ 𝑐 . 𝔐 Φ 𝑐 . 𝔐 Φ 𝑐 . 0 𝑐 . 0 0 Let 𝐷 , 𝐷 , … are disjoint sets in 𝒦. Since 𝔐 is measure relative to the λ– algebra 𝒦, 𝑗 1,2 Then, 𝔐 ⋃ 𝐷 ∑ 𝔐 𝐷 . So, we have 𝑐 𝔐 𝑐 𝔐 ⋃ 𝐷 𝑐 . 𝔐 ⋃ 𝐷 𝑐 . 𝔐 ⋃ 𝐷 𝑐 . ∑ 𝔐 𝐷 𝑐 . ∑ 𝔐 𝐷 ∑ 𝑐 . 𝔐 𝐷 ∑ 𝑐 . 𝔐 𝐷 ∑ 𝑐 . 𝔐 𝐷 𝑐 . 𝔐 𝐷 ∑ 𝑐 𝔐 𝑐 𝔐 𝐷 Hence, (𝔓 , 𝒦, 𝑐 𝔐 𝑐 𝔐 ) is measure space relative to the λ– algebra 𝒦. Now, we assume that (𝔓 , 𝒦, ∑ 𝑐 𝔐 ) is measure space relative to the λ– algebra 𝒦, when 𝑘 m and we prove this fact when 𝑘 m 1. Let (𝔓 , 𝒦, 𝔐 ) be a measure space relative to the λ– algebra 𝒦 and 𝑐 ∈ 0, ∞ for all 𝑗 1,2, … , 𝑚, 𝑚 1. Then ∑ 𝑐 𝔐 Φ ∑ 𝑐 𝔐 𝑐 𝔐 Φ ∑ 𝑐 . 𝔐 Φ 𝑐 . 𝔐 Φ 0 since, 𝔐 is measure relative to the λ– algebra 𝒦. Let 𝐷 , 𝐷 , … are disjoint sets in 𝒦. Since (𝔓 , 𝒦, ∑ 𝑐 𝔐 ) is measure space relative to the λ– algebra 𝒦, then ∑ 𝑐 𝔐 ⋃ 𝐷 ∑ ∑ 𝑐 𝔐 𝐷 . So, we have   79 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 ∑ 𝑐 𝔐 ⋃ 𝐷 ∑ 𝑐 𝔐 𝑐 𝔐 ⋃ 𝐷 ∑ 𝑐 . 𝔐 ⋃ 𝐷 𝑐 . 𝔐 ⋃ 𝐷 ∑ 𝑐 𝔐 ⋃ 𝐷 𝑐 . 𝔐 ⋃ 𝐷 ∑ ∑ 𝑐 𝔐 𝐷 𝑐 . ∑ 𝔐 𝐷 ∑ ∑ 𝑐 . 𝔐 𝐷 ∑ 𝑐 . 𝔐 𝐷 ∑ ∑ 𝑐 . 𝔐 𝐷 𝑐 . 𝔐 𝐷 ∑ ∑ 𝑐 𝔐 𝑐 𝔐 𝐷 ∑ ∑ 𝑐 𝔐 𝐷 . Hence, ∑ 𝑐 𝔐 is measure relative to 𝒦, therefore (𝔓 , 𝒦, ∑ 𝑐 𝔐 ) is measure space relative to the λ– algebra 𝒦. Definition 30 [1] A measure on a σ– field 𝒦 is a nonnegative, extended real-valued set function 𝔐 on 𝒦 such that whenever 𝐴 , 𝐴 , … form a finite or countably infinite collection of disjoint sets in 𝒦, we have, 𝔐 ⋃ 𝐴 ∑ 𝔐 𝐴 . Definition 31 [1, 3] A measure 𝔐 on a σ–field 𝒦 is said to be complete iff whenever A ϵ𝒦and 𝔐 𝐴 0, we have B ϵ𝒦for all 𝐵 ⊂ 𝐴. The following example shows that, if 𝔐 is a measure on σ–field 𝒦, then not necessarily that 𝔐 is complete. Example 32 Let 𝔓 ={1,2,3} and 𝒦 { Φ,{1},{2,3},𝔓 }. Then 𝒦 is σ–field of a set 𝔓 . If we define a set function 𝔐: 𝒦 → 0 , ∞ by 𝔐(𝐷) = 𝑜 ; 𝑖𝑓 𝐷 Φ 𝑜𝑟 𝐷 2,3 1 ; 𝑜𝑡ℎ𝑒𝑟 𝑤𝑖𝑠𝑒 Then 𝔐 is a measure on σ–field 𝒦, it is clear that 𝔐 is not complete, because {2,3}∈ 𝒦 and 𝔐 2,3 0 , now {2},{3}⊂{2,3}, but {2},{3}∉ 𝒦. Theorem 33 Every measure relative to the λ– algebra is complete. Proof Let 𝔐 be a measure relative to the λ– algebra 𝒦. Assume that A ϵ𝒦 such that 𝔐 𝐴 0, since 𝒦 is a λ– algebra, then B ϵ𝒦for all 𝐵 ⊂ 𝐴. Therefore 𝔐 is complete measure. Example 34 Let 𝔓 ={a,b,c,d} and 𝒦 { Φ,{a},{c},{d},{a,c},{c,d},{a,d},{a,c,d},𝔓 }. Then 𝒦 is λ– algebra of a set 𝔓 . If we define a set function 𝔐: 𝒦 → 0 , ∞ by 𝔐(𝐷) = 𝑜 ; 𝑖𝑓 𝐷 𝔓 1 ; 𝑖𝑓 𝐷 𝔓   80 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Then 𝔐 is a measure on λ– algebra 𝒦. Now, for any Aϵ𝒦 such that 𝔐 𝐴 0, then Bϵ𝒦 for all 𝐵 ⊂ 𝐴. Therefore 𝔐 is complete measure. 4. Conclusions The main results of this paper are the following: (1) Let 𝒦 ∈ be a collection of λ– algebra on 𝔓. Then ⋂ 𝒦∈ is a λ– algebra on 𝔓. (2) Let 𝒥 ⊆ P 𝔓 . Then λ 𝒥 is the smallest λ– algebra of 𝔓 which includes 𝒥. (3) Let 𝒥 ⊆ P 𝔓 . Then 𝒥 is a λ –algebra of a set 𝔓 if and only if 𝒥 λ 𝒥 . (4) Let 𝒥 ⊆ P 𝔓 and Φ 𝔇 ⊆ 𝔓. If 𝒦 is a λ –algebra of 𝔓 which includes 𝒥, then λ 𝒥 |𝔇 is a λ– algebra of a set 𝔇. (5) Let 𝒥 ⊆ P 𝔓 and Φ 𝔇 ⊆ 𝔓. Then λ 𝒥|𝔇 =λ 𝒥 |𝔇. (6) Every λ– algebra is a α– σ–field. (7) Every λ– algebra is a β– σ–field. (8) Every λ– algebra is a monotone class. 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