On studying bi-\(\Gamma\)-algebra and some related concepts

Document Type : Research Paper


1 Ministry of Education, General Directorate of Education in Diyala, Iraq

2 Department of Mathematics and Computer Applications, College of Science, Al-Nahraini University, Baghdad, Iraq

3 Department of Mathematics, College of Computer Science and Mathematics, Tikrit University, Tikrit, Salahaddin, Iraq


The aim of this research paper is to introduce the concept of bi-\(\Gamma\)-algebra space (bi-gamma algebra space). The concept of bi-\(\mu\)-measurable set in a bi-\(\Gamma\)-algebra space is defined. With this concept, some properties of bi-\(\Gamma\)-algebra space are proved. We then define various separation axioms for bi-\(\Gamma\)-algebra space such as \(M_{0},M_{1},\ M_{2},\ M_{3},\) and \(M_{4}\); then the relationships between them are studied. In addition, the concept of measurable function between two bi-measurable spaces is introduced and some results are discussed.


[1] M.M. Abbood, H.H. Ebrahim and A. Al-Fayadh, Study on Γ–algebra with some related concepts, Accepted to AIP Conf. Proc., 28/3/2021.
[2] V.I. Bogachev, Measure Theory I, Springer, 2007.
[3] R.M. Dudley, Real analysis and probability, Cambridge Univ. Press, 2002.
[4] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties, CRC Press. New York, 2015.
[5] P.R. Halmos, Measure Theory, Springer, 1976.
[6] B.A. Robert, Real Analysis and Probability, Academic Press, INC, University of Illinois, 1972.
[7] K. A. Ross and R. B. Ash, Real analysis and probability, Amer. Math. Month. 82(1) (1975) 91.
[8] Z. Wang and G. J. Klir, Generalized measure theory, Foundations, 25 (2008).[1] M.M. Abbood, H.H. Ebrahim and A. Al Fayadh, Study on Γ–algebra with some related concepts, Accepted to AIP Conf. Proc., 28/3/2021.
Volume 13, Issue 1
March 2022
Pages 1565-1572
  • Receive Date: 04 September 2021
  • Accept Date: 30 October 2021