Some Properties for the Restriction of P^*-field of Sets
Main Article Content
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.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
licenseTermsHow to Cite
Publication Dates
References
Wang, Z. ; Blir, G.J. Fuzzy Measure Theory. Springer Science and Business Media, LLC, New York, 1992; ISBN 978-1-4419-3225-9.
Ahmed, I.S. ; Ebrahim, H.H. Generalizations of σ-field and new collections of sets noted by δ-field, AIP Conf Proc. 2019, 2096, 1, 020019.
Robret, B. A. Real Analysis and Probability. Academic Press, Inc. New York. 1972.
Ahmed, I.S. ; Asaad, S.H. ; Ebrahim, H.H. Some new properties of an outer measure on a σ–field, Journal of Interdisciplinary Mathematics. 2021,24 (4), 947–952.
Wang, Z. ; George, J. ℬ. Generalized Measure Theory, 1st ed. Springer Science and Business Media, LLC, New York, 2009.
Endou, N. ; Naℬasho, ℬ. ; Shidama, Y. σ-ring and σ-algebra of Sets, Formaliz. Mathematics. 2015, 23 (1), 51–57.
Ebrahim, H.H.; Ahmed, I.S. 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. 2019,32 (2) , 62-70.
Ebrahim, H.H.; Rusul, A.A. λ–Algebra with Some Of Their Properties, Ibn Al Haitham Journal for Pure and Applied Science. 2020,33 (2) ,72-80.
Abbas, H.F. ; Ebrahim, H.H. ; Al-Fayadh, A. P^*–Field of sets and Some of its Properties, Accepted in Computers and Mathematics with Applications, 2022.