On Finitely Null-additive and Finitely Weakly Null-additive Relative to the σ–ring

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Samah H. Asaad
Ibrahim S. Ahmed
Hassan H. Ebrahim


     This article introduces the concept of finitely null-additive set function relative to the σ– ring and many properties of this concept have been discussed. Furthermore, to introduce and study the notion of finitely weakly null-additive set function relative to the σ– ring as a generalization of some concepts such as measure, countably additive, finitely additive, countably null-additive, countably weakly null-additive and finitely null-additive. As the first result, it has been proved that every finitely null-additive is a finitely weakly null-additive. Finally, the paper introduces a study of the concept of outer measure as a stronger form of finitely weakly null-additive.


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Asaad SH, Ahmed IS, Ebrahim HH. On Finitely Null-additive and Finitely Weakly Null-additive Relative to the σ–ring. Baghdad Sci.J [Internet]. 2022 Oct. 1 [cited 2022 Nov. 30];19(5):1148. Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/5771


Evans LC, Gariepy RF. Measure Theory and Fine Properties. 1st ed. CRC Press, New York. CRC Press, New York; 2015. 296 p

Ash RB. Real Analysis and Probability. 1st ed. Academic press, inc.(london) ltd.; 1972. 476 p.

Mackenzie A. A foundation for probabilistic beliefs with or without atoms. Theor Econ. 2019;14(2):709–78.

Wang P, Yu M, Li J. Monotone Measures Defined by Pan-Integral. Adv Pure Math. 2018;8(6):535–47.

Ahmed IS, Ebrahim HH. Generalizations of σ-field and new collections of sets noted by δ-field. AIP Conf Proc. 2019;2096(20019):(020019-1 )-(020019-6).

Endou N, Nakasho K, Shidama Y. σ-ring and σ-algebra of Sets. Formaliz Math. 2015;23(1):51–7.

Zhenyuan W, George JK. Generalized Measure Theory. 1st ed. Klir GJ, editor. Springer Science and Business Media, LLC. Springer Science and Business Media, LLC, New York; 2009. 381 p.

Gálvez-Rodríguez JF, Sánchez-Granero MÁ. The Distribution Function of a Probability Measure on a Linearly Ordered Topological Space. Mathematics. 2019;7(9):864.

Hanifat AF, Suryadi D, Sumiaty E. Reflective Inquiry: Why Area is Never Negative? J Phys Conf Ser. 2019;1280(4).

Nachbar J. Basic Lebesgue Measure Theory 1 1. Washington University in St. Louis; 2018. 1-28 p.

Dudzik D, Skrzyński M. An outer measure on a commutative ring. Algebr Discret Math. 2016;21(1):51–8.

Pap E. Extension of Null-Additive Set Functions on Algebra of Subsets. Novi Sad J Math. 2001;31(2):9–13.

Pap E. σ -Null-Additive Set Functions. Novi Sad J Math. 2002;32(1):47–57.

Li J, Mesiar R, Pap E. Atoms of weakly null-additive monotone measures and integrals. Inf Sci (Ny) [Internet]. 2014;257:183–92. Available from: http://dx.doi.org/10.1016/j.ins.2013.09.013