155 This work is licensed under a Creative Commons Attribution 4.0 International License. Some Properties for the Restriction of π“Ÿβˆ—β€“ 𝐟𝐒𝐞π₯𝐝 of Sets Abstract The restriction concept is a basic feature in the field of measure theory and has many important properties. This article introduces the notion of restriction of a non-empty class of subset of the power set on a nonempty subset of a universal set. Characterization and examples of the proposed concept are given, and several properties of restriction are investigated. Furthermore, the relation between the P*–field and the restriction of the P*–field is studied, explaining that the restriction of the P*–field is a P*–field too. In addition, it has been shown that the restriction of the P*–field is not necessarily contained in the P*–field, and the converse is true. We provide a necessary condition for the P*–field to obtain that the restriction of the P*–field is included in the P*–field. Finally, this article aims to study the restriction notion and give some propositions, lemmas, and theorems related to the proposed concept . Keywords: Οƒ βˆ’field, σ– ring, field, smallest Οƒ βˆ’field and restriction. 1. Introduction In the real analysis and probability, the σ–field concept is the class β„³ for a subset of a universal set 𝒰 such that 𝒰ϡℳ and it is closed under the complement, countable union [1] and [2]. The main reason for σ–field is the idea of measure, which is substantial in the real analysis as the basis of Lebesgue integrals, where it exponent as a family of events which may Ibn Al-Haitham Journal for Pure and Applied Sciences http://jih.uobaghdad.edu.iq/index.php/j/index: Journal homepage Doi: 10.30526/35.3.2814 Article history: Received 20 February 2022, Accepted 17 May, 2022, Published in July 2022. Hind F. Abbas Department of Mathematics / College of Computer Science and Mathematics / Tikrit University/ Iraq. hind.f.abbas35386@st.tu.edu.iq Ali Al-Fayadh Department of Mathematics and Computer Applications / College of Science/Al – Nahrain University/ Iraq aalfayadh@yahoo.com Hassan H. Ebrahim Department of Mathematics / College of Computer Science and Mathematics / Tikrit University/ Iraq hassan1962pl@tu.edu.iq https://creativecommons.org/licenses/by/4.0/ file:///F:/Ψ§Ω„ΨΉΨ―Ψ―%20Ψ§Ω„Ψ«Ψ§Ω†ΩŠ%202022/:%20http:/jih.uobaghdad.edu.iq/index.php/j/index mailto:hind.f.abbas35386@st.tu.edu.iq mailto:hind.f.abbas35386@st.tu.edu.iq mailto:aalfayadh@yahoo.com mailto:hassan1962pl@tu.edu.iq IHJPAS. 53 (3)2022 156 be assigned probability [3] and [4]. In the probability theory, a σ–field is essential in the conditional expected. Also, in statistics, sub σ–field is necessary for an official mathematical definition for sufficient statistic, where a statistic be a map or a random variable. A σ– ring idea was studied by [5] as a class β„³ such that B1\B2Ο΅β„³ and ⋃ Bn ∞ 𝑛=1 Ο΅β„³ wheneverB1, B2, … Ο΅β„³ . Many authors were interested in studying σ–field and σ– ring; for example, see [6], [7], and [8]. In this work, we denote a universal set by 𝒰. Preliminaries In the following, we mention some basic definitions and notations in measure space that will be used in this paper. Definition 2.1 [9]. Suppose β„³ is a class of subsets of 𝒰 . Then, β„³ is π‘‘β„Žπ‘’ π’«βˆ—β€“ field of 𝒰 if: 1- Ξ¦ Ο΅ β„³. 2- N, MΟ΅β„³; then, Nβ‹‚M Ο΅ β„³. 3- M2, … Ο΅ β„³; then, ⋃ Mi ∞ i=1 Ο΅ β„³. Example 2.2 [9]. Let 𝒰 ={1,2,3,4}. Consider β„³ ={ Ξ¦,{1},{1,2},{1,3},{1,2,3}}. Then β„³ is a π’«βˆ—β€“ field of 𝒰. Definition 2.3 [5]. The family of all subsets of 𝒰 is called a power set and denoted by P(𝒰), In symbols: P(𝒰) = { B ∢ B is a subset of 𝒰}. Proposition 2.4 [9] . If {β„³i}iϡΙ is a family of π’«βˆ—β€“ field of 𝒰, then so is β‹‚ β„³i iϡΙ . Definition 2.5 [9]. Let ℐ βŠ† P(𝒰). Then, π’«βˆ—(ℐ) = β‹‚{β„³i: β„³i is a 𝒫 βˆ—β€“ field of 𝒰 and β„³i βŠ‡ ℐ , βˆ€i ∈ Ξ™} is called the π’«βˆ—β€“ field generated by ℐ. Proposition 2.6 [9]. If ℐ βŠ† P(𝒰), then π’«βˆ—(ℐ) is the smallest π’«βˆ—β€“ field of 𝒰 that contains ℐ. Proposition 2.7 [5]. If β„³ is σ– field, then β„³ is a σ– ring. Proposition 2.8 [9]. Every σ– field is π’«βˆ—β€“ field. IHJPAS. 53 (3)2022 157 Proposition 2.9 [9]. Every σ– ring is π’«βˆ—β€“ field. 2. The Main Results In this section, the basic definitions and facts related to this work are recalled, starting with the following definition: Definition 3.1 Suppose β„³ is a π’«βˆ—β€“ field of 𝒰 and Ξ¦ β‰  ℬ βŠ† 𝒰, then a restriction of β„³ over ℬ is defined as: β„³|ℬ = { N: N =Mβ‹‚ ℬ, for some M Ο΅ β„³}. Proposition 3.2 Suppose β„³ is π’«βˆ—β€“ field of 𝒰 and Ξ¦ β‰  ℬ βŠ† 𝒰, then β„³|ℬ is 𝒫 βˆ—β€“ field on ℬ. Proof. Since Φϡ β„³ and Ξ¦ = Φ⋂ℬ, then Φϡ β„³|ℬ . Let N1, N2Ο΅ β„³|ℬ, then there is M1, M2Ο΅ β„³ such that Ni=Mi⋂ℬ where i=1,2 which implies that N1β‹‚N2= ( M1⋂ℬ) β‹‚(M2⋂ℬ ) = (M1β‹‚M2)⋂ℬ. Since β„³ is a π’«βˆ—β€“ field of 𝒰, then, M1β‹‚M2Ο΅ β„³. Thus N1β‹‚N2Ο΅ β„³|ℬ Let N1, N2, … Ο΅ β„³|ℬ, then there is M1, M2, … Ο΅ β„³ such that Ni=Mi⋂ℬ where i=1, 2… which implies that ⋃ , Ni ∞ i=1 = ⋃ (Mi ∞ i=1 β‹‚ ℬ)= (⋃ Mi ∞ i=1 ) β‹‚ ℬ.. Since β„³ is a π’«βˆ—β€“ field of a set 𝒰, then ⋃ Mi ∞ i=1 Ο΅ β„³ and hence ⋃ Ni ∞ i=1 Ο΅ β„³|ℬ. Thus, β„³|ℬis a 𝒫 βˆ—β€“ field on ℬ. Proposition 3.3 If β„³ is π’«βˆ—β€“ field of 𝒰 and C βŠ† ℬ βŠ† 𝒰 such that CΟ΅β„³, then CΟ΅β„³|ℬ. Proof. Clearly. The following examples explain that if β„³ is a π’«βˆ—β€“ field of a set 𝒰 , then it is not necessarily that : 1- β„³|ℬ βŠ† β„³. 2- β„³ βŠ† β„³|ℬ Example 3.4 Let 𝒰 ={1,2,3,4}and β„³ ={ Ξ¦,{1,3},{1,2,3},{1,3,4},𝒰 }. Then, β„³ is a π’«βˆ—β€“ field of 𝒰. If ℬ ={2,3,4}, then β„³|ℬ ={ Ξ¦,{3},{2,3},{3,4}, ℬ}. It is clear that β„³|ℬ ⊈ β„³, since {3}∈ β„³|ℬ but {3}βˆ‰ β„³. Example 3.5 Let 𝒰 ={1,2,3,4}and β„³ ={ Ξ¦,{1,2},{1,2,3},{1,2,4},𝒰 }. Then, β„³ is a π’«βˆ—β€“ field of 𝒰. If ℬ ={2,3,4}, then β„³|ℬ ={ Ξ¦,{2},{2,3},{2,4}, ℬ}. It is clear that β„³ ⊈ β„³|ℬ, since {1, 2}∈ β„³ but {1,2}βˆ‰ β„³|ℬ. IHJPAS. 53 (3)2022 158 Proposition 3.6 If β„³ is π’«βˆ—β€“ field on 𝒰 and Ξ¦ β‰  ℬ βŠ† 𝒰 such that ℬϡ β„³. Then β„³|ℬ= {C βŠ† ℬ: CΟ΅ β„³}. Proof. Assume that NΟ΅ β„³|ℬ, then N=Mβ‹‚ ℬ, for some MΟ΅ β„³ and thus NΟ΅ β„³. Hence, NΟ΅{C βŠ† ℬ: CΟ΅ β„³}. Therefore, β„³|ℬ βŠ† {C βŠ† ℬ: CΟ΅ β„³}. Let DΟ΅ {C βŠ† ℬ : CΟ΅ β„³}. Then D βŠ† ℬ and D Ο΅ β„³, hence D = Dβ‹‚ ℬ, but DΟ΅ β„³, then DΟ΅ β„³|ℬ. So, we get {C βŠ† ℬ : C Ο΅ β„³}βŠ† β„³|ℬ. Consequentially, β„³|ℬ={C βŠ† ℬ : CΟ΅ β„³}. Corollary 3.7 If β„³ is π’«βˆ—β€“ field on 𝒰 and Ξ¦ β‰  ℬ βŠ† 𝒰 such that ℬϡ β„³. Then, β„³|ℬ βŠ† β„³. Proof. The proof follows Proposition 3.6. Definition 3.8 If 𝒰 is a universal set and ℐ βŠ† P(𝒰) and Ξ¦ β‰  ℬ βŠ† 𝒰, then a restriction of ℐ on ℬ is defined as: ℐ|ℬ = {N: N=Mβ‹‚ ℬ, for some MΟ΅ ℐ}. Proposition 3.9 If ℐ βŠ† P(𝒰) and Ξ¦ β‰  ℬ βŠ† 𝒰. Assume β„³ is a π’«βˆ—β€“ field of 𝒰 that contains ℐ and ℬϡ β„³, then π’«βˆ—(ℐ)|ℬ is a 𝒫 βˆ—β€“ field of ℬ. Proof. The proof is done by proposition 2.6 and 3.2 Theorem 3.10 Assume ℐ βŠ† P(𝒰) and Ξ¦ β‰  ℬ βŠ† 𝒰, then π’«βˆ— (ℐ|ℬ ) is the smallest 𝒫 βˆ—β€“ field on ℬ that contain ℐ|ℬ , where π’«βˆ— (ℐ| ℬ ) = β‹‚{β„³i|ℬ : β„³i|ℬ is a 𝒫 βˆ—β€“ field of ℬ and β„³i|ℬ βŠ‡ ℐ|ℬ, βˆ€i ∈ Ξ™}. Proof. In the same way as in proposition 2.4, we can prove that π’«βˆ—(ℐ| ℬ ) is a π’«βˆ—β€“ field on ℬ. To prove that π’«βˆ—(ℐ|ℬ ) βŠ‡ ℐ|ℬ , assume that β„³i|ℬ is a 𝒫 βˆ—β€“ field on ℬ and β„³i|ℬ βŠ‡ ℐ|ℬ , βˆ€i ∈ Ξ™, then ℐ|ℬ βŠ† β‹‚ β„³i|ℬ iβˆˆΞ™ ; hence ℐ|ℬ βŠ† 𝒫 βˆ—(ℐ|ℬ ). Now, let β„³ βˆ—|ℬ be a 𝒫 βˆ—β€“ field on ℬ such that β„³ βˆ—|ℬ βŠ‡ ℐ|ℬ. Then, β„³ βˆ—|ℬ βŠ‡ 𝒫 βˆ—(ℐ|ℬ ). Therefore, 𝒫(ℐ|ℬ ) is the smallest 𝒫 βˆ—β€“ field on ℬ containing ℐ|ℬ . Theorem 3.11 If ℐ βŠ† P(𝒰) and Ξ¦ β‰  ℬ βŠ† 𝒰, define a class β„³ by: β„³ ={M βŠ† 𝒰 : Mβ‹‚ ℬ Ο΅ π’«βˆ—(ℐ|ℬ ) }. Then β„³ is a 𝒫 βˆ—β€“ field on a set 𝒰. IHJPAS. 53 (3)2022 159 Proof. By Theorem 3.10, we have π’«βˆ—(ℐ| ℬ ) as a π’«βˆ—β€“ field on ℬ, so Ξ¦ Ο΅ π’«βˆ—(ℐ|ℬ ). Since Ξ¦= Ξ¦ β‹‚ ℬ, then we get Φϡℳ. Assume that M1, M2 Ο΅ β„³. Then (Mi⋂ℬ)Ο΅ 𝒫 βˆ—(ℐ|ℬ ), for each i=1,2. Now, (M1β‹‚M2)⋂ℬ = ( M1⋂ℬ) β‹‚(M2⋂ℬ ). Since 𝒫 βˆ—(ℐ| ℬ ) } is a π’«βˆ—β€“ field on ℬ, then ( M1⋂ℬ) β‹‚(M2⋂ℬ )ϡ𝒫 βˆ—(ℐ|ℬ ) and hence (M1β‹‚M2)⋂ℬϡ𝒫 βˆ—(ℐ|ℬ ), thus M1β‹‚M2Ο΅ β„³. Let M1, M2 , … Ο΅ β„³. Then (Mi⋂ℬ)Ο΅ 𝒫 βˆ—(ℐ|ℬ ), for i=1,2,… Since π’«βˆ—(ℐ| ℬ ) } is π’«βˆ—β€“ field on ℬ, then ⋃ (Mi ∞ i=1 β‹‚ ℬ)ϡ𝒫 βˆ—(ℐ|ℬ ). Now, (⋃ Mi ∞ i=1 )⋂ℬ = ⋃ (Mi ∞ i=1 β‹‚ ℬ)ϡ𝒫 βˆ—(ℐ|ℬ ), thus ⋃ Mi ∞ i=1 Ο΅ β„³. Therefore, β„³ is π’«βˆ—β€“ field on a universal set 𝒰. Theorem 3.12 If 𝒰 is a universal set and ℐ βŠ† P(𝒰) such that Ξ¦ β‰  ℬ βŠ† 𝒰, then π’«βˆ—(ℐ| ℬ ) = π’«βˆ—(ℐ)| ℬ . Proof. By proposition 2.6, we have π’«βˆ—(ℐ) is π’«βˆ—β€“ field on 𝒰. So, we get π’«βˆ—(ℐ)|ℬ is π‘Ž 𝒫 βˆ—β€“ field on ℬ by proposition 3.2. Assumeπ‘‘β„Žπ‘Žπ‘‘ Nϡℐ|ℬ. Then N = Mβ‹‚ ℬ for some MΟ΅ ℐ. But ℐ βŠ† π’«βˆ—(ℐ), so we have MΟ΅ π’«βˆ—(ℐ) and thus NΟ΅ π’«βˆ—(ℐ)|ℬ. Hence ℐ|ℬ βŠ† 𝒫 βˆ—(ℐ)|ℬ. Therefore, 𝒫 βˆ—(ℐ)|ℬ is a 𝒫 βˆ—β€“ field on ℬ that containing ℐ|ℬ . By Theorem 3.10, we have π’«βˆ—(ℐ| ℬ ) is the smallest π’«βˆ—β€“ field on ℬ that containing ℐ|ℬ, which implies that π’«βˆ—(ℐ| ℬ ) βŠ† π’«βˆ—(ℐ)|ℬ. Now, if we define a class β„³ by β„³= {C βŠ† 𝒰 : Cβ‹‚ ℬ Ο΅ π’«βˆ—(ℐ|ℬ ) }, then in Theorem 3.11, we have β„³ as π‘Ž 𝒫 βˆ—β€“ field on 𝒰. Let CΟ΅ ℐ, then (C ∩ ℬ) ϡℐ|ℬ, but ℐ|ℬ βŠ† 𝒫(ℐ|ℬ ) implies that (C ∩ ℬ)Ο΅ 𝒫 βˆ—(ℐ|ℬ ), hence CΟ΅ β„³ and ℐ βŠ† β„³. Now, if we assume that NΟ΅ π’«βˆ—(ℐ)|ℬ, then N= M ∩ ℬ, for some MΟ΅ 𝒫(ℐ). But 𝒫 βˆ—(ℐ) βŠ† β„³, then MΟ΅ β„³, hence NΟ΅ π’«βˆ—(ℐ|ℬ ). Consequentially, 𝒫 βˆ—(ℐ)|ℬ βŠ† 𝒫 βˆ—(ℐ| ℬ ). This completes the proof. 3. Conclusions We tried to define the concept of measure relative to the π’«βˆ—β€“ field β„³ of 𝒰 and also define the idea of the restriction of measure on β„³|ℬ of a set ℬ. Also, we discuss many properties of these notions. In this article, the idea of π’«βˆ—β€“ field is given to refer to the generalization of each σ– field and σ–ring. Furthermore, some properties of the purposed notion are proven as explained below: 1. Let β„³ be a π’«βˆ—β€“ field of a set 𝒰 and let ℬ be a nonempty subset of 𝒰. Then, β„³|ℬ is a π’«βˆ—β€“ field of a set ℬ. 2. Assume π‘‘β„Žπ‘Žπ‘‘ β„³ is a π’«βˆ—β€“ field on 𝒰 and A βŠ† ℬ βŠ† 𝒰. If AΟ΅ β„³, then AΟ΅ β„³|ℬ. 3. If β„³ is a π’«βˆ—β€“ field and ℬ be a nonempty subset of 𝒰 such that ℬϡ β„³. Then β„³|ℬ= {A βŠ† ℬ: AΟ΅ β„³}. 4. Suppose that β„³ is a π’«βˆ—β€“ field and ℬ βŠ† 𝒰 such that ℬϡ β„³. Then β„³|ℬ βŠ† β„³. 5. 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