Sains Malaysiana 46(1)(2017): 175–179

http://dx.doi.org/10.17576/jsm-2017-4601-22

Abstract Characterization of a Conditional Expectation Operator on the Space of

Measurable Sections

(Pencirian Abstrak Bagi Pengendali Jangkaan Bersyarat di Ruang Bahagian yang

Boleh Diukur)

INOMJON GANIEV* & TORLA HASSAN

Department of Science in Engineering, Faculty of Engineering, International Islamic University Malaysia, P.O. Box 10, 50728 Kuala-Lumpur, Malaysia

Diserahkan: 6 Mac 2016/Diterima: 22 April 2016

ABSTRACT

A conditional expectation operator plays an important role in geometry of Banach spaces. However, the main issue is with regards to the existence of a conditional expectation operator that permits other objects to be considered such as martingales and martingale convergence theorems. Thus, the purpose of this study is to provide an abstract characterization of a conditional expectation operator on a space of measurable sections.

Keywords: Abstract characterization; conditional expectation operator; measurable section

ABSTRAK

Pengendalian jangkaan bersyarat memainkan peranan yang penting di dalam geometri ruang Banach. Walau bagaimanapun, isu utama adalah berkaitan dengan kewujudan pengendali jangkaan bersyarat yang membenarkan objek lain yang perlu dipertimbangkan seperti teori penumpuan martingale dan martingale. Dengan itu, tujuan kajian ini adalah untuk memberikan pencirian abstrak bagi pengendali jangkaan bersyarat di ruang bahagian yang boleh diukur.

Kata kunci: Bahagian yang boleh diukur; pencirian abstrak; pengendali jangkaan bersyarat

RUJUKAN

Ganiev, I.G. & Shahidi, F.A. 2015. Conditional expectation operator on the space of measurable sections. AIP Conference Proceedings 1660: 090046. doi: 10.1063/1.4926635.

Ganiev, I.G & Gharib, S.M. 2013. The Bochner integral for measurable sections and its properties. Annals of Functional Analysis 4(1): 1-10.

Ganiev, I.G. 2006. Measurable bundles of lattices and their application. In Studies on Functional Analysis and its Applications. Nauka, Moscow (Russian). pp. 9-49.

Gutman, A.E. 1995. Banach bundles in lattice normed spaces. In Linear Operators, Associated with Order. Novosibirsk: Izd. IM SO RAN. pp. 63-211 (Russian).

Douglas, R.G. 1965. Contractive projections on an space. Pacific Journal of Mathematics 15(2): 443-462.

Diestel, J. & Uhl, J.J. 1977. Vector Measures. American Mathematical Society, Providence, R.I. Kusraev, A.G. 1985. Vector Duality and Its Applications. Novosibirsk, Nauka (Russian).

Kusraev, A.G. 2000. Dominated operators. Mathematics and and its Applications. Dordrecht: Kluwer Academic Publishers.

Landers, D. & Rogge, L. 1981. Characterization of conditional expectation operators for Banach-valued functions. Proc. Amer. Math. Soc. 81(1): 107-110.

Rao, M.M. 1976. Two characterizations of conditional probability. Proc. Amer. Math. Soc. 19: 75-80.

Rao, M.M. 1965. Conditional expectations and closed projections. Nederl. Akad. Wetensch. Proc. Ser. A 68 Indag. Math. 27: 100-112.

Vakhania, N.N., Tarieladze, V.I. & Chobanyan, S.A. 1987. Probability distributions on Banach spaces. In Mathematics and Its Applications, vol. 14, Reidel, D. (ed), Publishing Co., Dordrecht.

*Pengarang untuk surat-menyurat; email: ganiev1@rambler.ru