26 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 Abstract The objective of this paper is, first, study a new collection of sets such as –field and we discuss the properties of this collection. Second, introduce a new concepts related to the –field such as measure on –field, outer measure on –field and we obtain some important results deals with these concepts. Third, introduce the concept of null-additive on –field as a generalization of the concept of measure on –field. Furthermore, we establish new concept related to - field noted by weakly null-additive on –field as a generalizations of the concepts of measure on and null-additive. Finally, we introduce the restriction of a set function on –field and many of its properties and characterizations are given. Keywords: –field, measure on –field, monotone measure, null-additive. 1. Introduction The theory of measure is an important subject in mathematics. In 1972, Robret [1], discusses many details about measure and proves some important results in measure theory. The notion of –field was studied by Robret and Dietmar, where be a nonempty set. A collection is said to –field iff and is closed under complementation and countable union [1, 2]. Zhenyuan and George in 2009 and Junhi, Radko and Endre in 2014 are used the concept of null-additive on –field, where be a –field, then a set function , - is called null-additive on if are disjoint sets in and ( ) , then ( ) ( ) [3,4]. In 2016, Juha used the concept of –field to define measure, where be a –field, then a measure on is a set function , - such that ( ) and if form a finite or countably infinite collection of disjoint sets in , then ( ) ∑ ( ) [5]. and also used power set to define outer measure, where be a non-empty set, then a set function ( ) , - is called outer measure, if ( ) and if such that , then ( ) ( ) and if are subsets of , then ( ) ∑ ( ) [5]. The concept of monotone measure was studied by Peipe, Minhao and Jun in 2018, where be a –field, then a set function , - is called monotone measure, if ( ) and if such that , then ( ) ( ) [6]. On a New Kind of Collection of Subsets Noted by 𝛅–field and Some Concepts Defined on 𝛅–field Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Hassan H. Ebrahim hassan1962pl@tu.edu.iq Ibrahim S. Ahmed Article history: Received 27 January 2019, Accepted 13 March 2019, Publish May 2019 10.30526/32.2.2140 Doi: Department of Mathematics, College of Computer science and Mathematics, University of Tikrit, Tikrit, Iraq. ibrahimsalhahmed69@gmail.com mailto:hassan1962pl@tu.edu.iq mailto:hassan1962pl@tu.edu.iq mailto:ibrahimsalhahmed69@gmail.com mailto:ibrahimsalhahmed69@gmail.com 26 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 The main aim of this paper is to introduce and study new concepts such as –field, measure on –field, outer measure on –field and null-additive on –field and we give basic properties, characterizations and examples of these concepts. 2. The Main Results Let be a nonempty set. Then a collection of all subsets of a set , denoted by ( ), and it's called a power set of Definition 1 Let be a nonempty set. A collection ( ) is said to be – of a set if the following conditions are satisfied: 1- . 2- If is a nonempty set in and , then . 3- If , then ⋂ . Proposition 2 For any – of a set , the following hold: 1- 2- If , then ⋂ 3- If , then ⋂ . 4- If , then . 5- , then . Proof It is easy, so we omitted. Example 3 Let ={1, 2, 3, 4} and { , {1,2},{1,2,3}, {1,2,4}, }. Then is a – of a set . Definition 4 Let be a nonempty set and is a – of a set .Then a pair ( , ) is called measurable space and any member of is called a measurable set. Proposition 5 Let * + be a sequence of – of a set . Then ⋂ is a – of a set . Proof Since is – , then , , hence and ⋂ , therefore ⋂ . Let ⋂ such that , then , but So, we get , hence ⋂ . Let ⋂ . Then , and ⋂ , which is implies that ⋂ ⋂ . Hence ⋂ is a – . Definition 6 Let be a – of a set and let be a non-empty subset of . Then the restriction of on is denoted by and define as: ={ : = ⋂ , for some }. Proposition 7 Let be a – of a set and be a non-empty subset of such that . Then = { : }. 26 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 Proof Let . Then = ⋂ , for some , hence . Therefore { : } and { : }. Let { : }. Then and , hence = ⋂ , but , then which is implies that{ : } , therefore ={ : }. Corollary 8 Let be a – of a set and a non-empty subset of such that . Then . Proof From Proposition 7, we have ={ : }. Now, for any , then { : }. Hence and , therefore . Proposition 9 Let be a – of a set and let be a non-empty subset of such that . Then is a - field of a set . Proof Since is a – of , then . Since , then ⋂ and Since ⋂ , then Φ Let such that Then . But and is a – of a set , then . Now, and , then Let Then there exist such that = ⋂ where i=1,2,…, now ⋂ =(⋂ ) ⋂ But, is a – , then ⋂ . Hence ⋂ .Therefore is a – of a set . If we take Example 3 and if we assume that ={1,2,4}, then ={ ,{1, 2}, } is a – of a set and . Definition 10 Let be a – of a set . A measure on is a set function , - such that ( ) and if form a finite or countably infinite collection of disjoint sets in , then ( ) ∑ ( ). Example 11 Let be a – of a set and define , - by ( ) = 0, for all . Then is a measure on . A measure space is a triple ( ) where is a nonempty set and is a – of a set and is a measure on Definition 12 Let be a – of a set . A countably subadditive on is a set function , - such that ( ) ∑ ( ) where and . If this requirement holds only for finite collection of disjoint sets in , then is said to be finitely subadditive on a – . Definition 13 Let be a – of a set . Then a set function , - is said to be monotone measure, if it satisfies the following requirements: 1- ( ) 2- If and , then ( ) ( ). 26 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 Definition 14 Let be a – of a set . Then a set function , - is called outer measure, if it satisfies the following requirements: 1- ( ) . 2- If and , then ( ) ( ). 3- If , then ( ) ∑ ( ). Lemma 15 Let be an outer measure on – of a set and , ). If : , - is defined by ( )( ) ( ) , then ( ) is an outer measure on . Proof Since is an outer measure on and , then ( ) =0 and ( )( ) = 0. Let and , then and ( ) ( ). Since ( )( ) ( ) ( ) ( )( ) Let , then So, we have ( )( ) ( ) ∑ ( ) But, ∑ ( ) ∑ ( ) ∑ ( )( ). Therefore is an outer measure on . Lemma 16 Let and be two outer measures on a – of a set . If : , - is defined by ( )( ) ( ) ( ), , then is an outer measure on . Proof Since and are outer measure on – and , then ( ) = ( ) = 0 and ( )( ) = 0. Let and , then and ( ) ( ) and ( ) ( ). So we have, ( )( ) ( ) ( ) ( ) + ( ) ( )( ) Let , then . So, we have ( )( ) ( ) ( ) ∑ ( ) ∑ ( ) ∑ , ( ) ( )- ∑ ( )( ). Therefore is an outer measure on . The proof of the following proposition consequence from Lemma (15 and 16) with mathematical induction. Proposition 17 Let , ,…, be outer measure on a – of a set and , ) for all . If a set function ∑ : , - is defined by: (∑ ) ( ) ∑ ( ) , then ∑ is an outer measure on – . Proof Since , ) and is an outer measure on a – for all . Then by Lemma15 we get is an outer measure on a – = 22 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 Let . Then we prove that (∑ ) is an outer measure on by mathematical induction. If , then is an outer measure on by Lemma16. Suppose that (∑ ) is an outer measure on , then we must prove that (∑ ) is an outer measure on , whenever is an outer measure on . (∑ ) ( ) (∑ )( ) (∑ )( ) ( ) since (∑ ) and are outer measure on Let and . Then (∑ ) ( ) (∑ ) ( ) and ( ) ( ) (∑ ) ( ) (∑ ) ( ) ( ) (∑ ) ( ) ( )since (∑ ) and are outer measure (∑ )( ) (∑ )( ) Let . Then (∑ ) ( ) (∑ ) ( ) (∑ ) ( ) ( ) ∑ (∑ ) ( ) ∑ ( ) ∑ ,(∑ ) ( ) ( )- ∑ (∑ )( ) ∑ (∑ )( ). Therefore, ∑ is an outer measure on Definition 18 Let be a – of a set . Then a set function , - is called null-additive on iff are disjoint sets in and ( ) , then ( ) ( ) . Example 19 Let = {1,2} and = { , {1} , {2}, } and define , - by: ( ) = { . Then is a null-additive. Proposition 20 Let be a – of a set . Then every measure is null-additive. Proof Let be a measure on – and let are disjoint sets in and ( ) . Then ( ) ( ) + ( ) ( ). Hence is a null-additive . While the converse is not true and Example 19 indicate that is null-additive but not measure, because {1},{2} are disjoint sets in but (* + * +) (* +) + (* +). Lemma 21 Let be a null-additive on a – of a set and ( ). If : , - is defined by: ( )( ) ( ) , then ( ) is a null-additive on . Proof Let be disjoint sets in such that ( )( ) . Then ( ) and hence ( ) since . Now, ( )( ) ( ) ( ) ( )( ) Therefore, is a null-additive on . 26 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 Lemma 22 Let and be two null-additives on a – of a set . If : , - is defined by: ( )( ) ( ) ( ) , then is a null-additive on . Proof Let be disjoint sets in such that( )( ) . Then ( ) ( ) , hence ( ) ( ) since and are null-additive on . Now, ( )( ) ( ) + ( ) ( ) + ( ) ( + )( ) Therefore, is a null-additive on . Proposition 23 Let , ,…, be a null-additive on a – of a set and ( ) for all . If a set function ∑ : , - is defined by: (∑ ) ( ) ∑ ( ) , then ∑ is a null-additive on . Proof Since ( ) and is null-additive on for all , then by Lemma 21, we get is a null-additive on . Let If , then is a null-additive on by Lemma 22. Let are disjoint sets in such that(∑ ) ( ) . Then ( ) for all . (∑ ) ( ) = ( ) ( ) = ( ) ( ) since is a null-additive and ( ) , = (∑ ) ( ). Hence ∑ is a null-additive on Definition 24 Let be a – of a set and let [0, ] be a set function and . If : [0, ] is define by ( ) = ( ) for all , then is called – restriction of Proposition 25 Let be a – of a set and . If is a measure on , then: (1) is a measure on . (2) ( ) = ( ), whenever . (3) ( ) = 0, whenever are disjoint sets in . Proof (1). Since is a – , then and ( ) = 0. From definition of we get, ( )= ( ) = ( ) = 0. Let are disjoint sets in , then . Since n=1,2,…, then and hence ( ) . So, we have ( ) = (( ) ) = ( ( )) = ∑ ( ) = ∑ ( ) . Therefore, is a measure on (2). Since , then = . So, we have ( ) = ( ) = ( ) (3). Since are disjoint sets in , then = and ( ) = ( ) = ( ) = 0. 26 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 Proposition 26 Let be a – of a set and . If is an outer measure on , then is an outer measure on Proof Since is a – , then and ( ) = 0. From definition of we get , ( ) = ( ⋂ ) = ( ) = 0. Let and , then ⋂ ⋂ and each of ⋂ ⋂ . Since is an outer measure on , then ( ⋂ ) ( ⋂ ) .So, we have ( ) ( ). Let . Then and n=1,2,…, hence ( ) . So, we have, ( ) = (( ) ) = ( ( )) ∑ ( ) = ∑ ( ) Therefore, is an outer measure on From Proposition 26, we conclude that if is a monotone measure on , then is a monotone measure on , where is a – of a set and . Proposition 27 Let be a – of and . If is a null-additive on , then is a null-additive on Proof Let be disjoint sets in and ( ) . Then ( ⋂ ) Now, ( ) = (, -⋂ ) = (, ⋂ - , ⋂ -) = ( ⋂ ) since is a null-additive on = ( ) by definition of Hence, is a null-additive on Proposition 28 Let be a – of and . If is a measure on , then is a null-additive on Proof It is easy, so we omitted. Definition 29 Let be a – of a set and [0, ] be a set function and be a non-empty subsets of such that . If : [0, ] is define by: ( ) = ( ) for all , then is called the restriction of on Proposition 30 Let be a measure on – of a set and such that . Then is a measure on a – of a set . Proof Since is a – of a set , then and ( ) = 0. Since , then by definition of , we get ( ) = ( )= 0. Let be disjoint sets in . Then and for all n=1,2,…, hence . So, we have ( ) = ( ) = ∑ ( ) since is a measure on = ∑ ( ) Therefore, is a measure on a – of a set . 26 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 If is an outer measure on – of a set , then we need the following two facts to prove that is an outer measure on a – of a set . Lemma 31 Let be a monotone measure on – of a set and such that . Then is a monotone measure on a – of a set . Proof Let be a monotone measure on , then ( ) = 0. Since is a – , then . From definition of , we get ( ) = ( )= 0. Let such that , then and . Since is a monotone measure on , then ( ) ( ). But , then ( )= ( ) and ( ) = ( ), hence ( ) ( ) and is monotone measure on of . Lemma 32 Let be a countably subadditive on – of a set and such that , then is a countably subadditive on a – of a set Proof Let and ,then and . Since be a countably subadditive on , then ( ) ∑ ( ), but . So, we have ( ) ( ) and ( ) ( ) for all n=1,2,…, hence ( ) ∑ ( ) and is a countably subadditive on of a set . Proposition 33 Let be an outer measure on – of a set and such that . Then is an outer measure on – of a set . Proof Since is an outer measure on , then is a monotone measure and countably subadditive. By Lemma 31 and Lemma 32 we have is a monotone measure and countably subadditive on of . Therefore is an outer measure on of . Proposition 34 Let be a null-additive on – of a set and such that . Then is a null-additive on – . Proof: Let be disjoint sets in and ( ) . Then ( ) Now, ( ) = (, ) = ( ) since is a null-additive on = ( ) by definition of . Hence, is a null-additive on 3. Conclusions The main results of this paper are the following: (1) Let be a nonempty set. A collection ( ) is said to be – of a set if the following conditions are satisfied: 1. . 2. If is a nonempty set in and , then . 3. If , then ⋂ . (2) Let * + be a sequence of – of a set . Then ⋂ is a – of a set . 67 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 (3) Let be a – of a set and let be a non-empty subset of . Then the restriction of on is denoted by and ={ : = ⋂ , for some }. (4) Let be a – of a set . Then every measure is null-additive. (5) Let , ,…, be null-additive on a – of a set and ( ) for all . If a set function ∑ : , - is defined by: (∑ ) ( ) ∑ ( ) , then ∑ is a null-additive on . (6) Let be a – of a set and . If is a measure on , then: 1. is a measure on . 2. ( ) = ( ), whenever . 3. ( ) = 0, whenever are disjoint sets in . (7) Let be a – of a set and . If is an outer measure on , then is an outer measure on (8) Let be a – of and . If is a null-additive on , then is a null-additive on (9) Let be a measure on – of a set and such that . Then is a measure on a – of a set . (10) Let be a monotone measure on – of a set and such that . Then is a monotone measure on a – of a set . References 1. Robret, B.A. Real Analysis and Probability, Academic Press, Inc, New York.1972, 4-16. 2. Dietmar, A.S. Measure and Integration, ETH Zürich [Internet].2016.Available from: https://people.math.ethz.ch/~salamon/PREPRINTS/measure.pdf. 3. Zhenyuan, W.; George, J.K. Generalized Measure Theory, Springer Science and Business Media, LLC. 2009, 133-134. 4. Jun, Li.; Radko, M.; Endre, P. Atoms of weakly null-additive monotone measures and integrals. Information Sciences. elsevier Inc. 2014, 257, 183–192, doi: 10.1016/j.ins.2013.09.013. 5. Juha, K. measure and Integrals. Aalto Math [internet].2016.Available form: https://math.aalto.fi/~jkkinnun/files/measure_and_integral.pdf. 6. Peipei, W.; Minhao, Yu.; Jun, Li. Monotone Measures Defined by Pan-Integral. Advances in Pure Mathematics. 2018, 8, 535–547, doi:10.4236/apm.2018.86031. https://math.aalto.fi/~jkkinnun/files/measure_and_integral.pdf https://math.aalto.fi/~jkkinnun/files/measure_and_integral.pdf