Abstract
The central purpose of this paper is to illustrate that combining the recently developed theory of random conjugate spaces and the deep theory of Banach spaces can, indeed, solve some difficult measurability problems which occur in the recent study of the Lebesgue (or more general, Orlicz)-Bochner function spaces as well as in a slightly different way in the study of the random functional analysis but for which the measurable selection theorems currently available are not applicable. It is important that this paper provides a new method of studying a large class of the measurability problems, namely first converting the measurability problems to the abstract existence problems in the random metric theory and then combining the random metric theory and the relative theory of classical spaces so that the measurability problems can be eventually solved. The new method is based on the deep development of the random metric theory as well as on the subtle combination of the random metric theory with classical space theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Wang Z K. Introduction to random functional analysis. Adv Math Sinica, 5(1): 45–71 (1962) (in Chinese)
Yan J A. Lecture on Measure Theory. 2nd Ed. Beijing: Science Press, 2004 (in Chinese)
Bharucha-Reid, A T. Fixed point theorems in probabilistic analysis. Bull Amer Math Soc, 82: 641–657 (1976)
Wagner D H. Survey of measurable selection theorems. SIAM J Control and Optimization, 15(5): 859–903 (1977)
Guo T X. Some basic theories of random normed linear spaces and random inner product spaces. Acta Anal Funct Appl, 1(2): 160–184 (1999)
Guo T X. Survey of recent developments of random metric theory and its applications in China (I). Acta Anal Funct Appl, 3(2): 129–158 (2001)
Guo T X. Survey of recent developments of random metric theory and its applications in China (II). Acta Anal Funct Appl, 3(3): 208–230 (2001)
Guo T X. The Radon-Nikodým property of conjugate spaces and the ω*-µ-equivalence theorem for the ω*-µ-measurable functions. Sci China Ser A: Math, 39(10): 1034–1041 (1996)
Guo T X. Representation theorems of the dual of Lebesgue-Bochner function spaces. Sci China Ser A: Math, 43(3): 234–243 (1999)
Hu Z B, Lin B L. Extremal structure of the unit ball of L p(µ,X)*. J Math Anal Appl, 200: 567–590 (1996)
Guo T X, Li S B. The James theorem in complete random normed modules. J Math Anal Appl, 308: 257–265 (2005)
Dunford N, Schwartz J T. Linear Operators, Part I. New York: Interscience, 1957
Guo T X. The theory of probabilistic metric spaces with applications to random functional analysis. Master’s Thesis. Xi’an: Xi’an Jiaotong University, 1989
Guo T X. The theory of random normed modules and its applications. In: Liu P D, ed. Proceedings of the International Conference and 13th Academic Symposium in China on Functional Space Theory and Its Applications. London: Research Information Ltd UK, 2004, 57–66
Guo T X. A new approach to random functional analysis. In: Feng E B, ed. Proceedings of China First Postdoctoral Academic Conference (II). Beijing: National Defence Industry Press, 1993, 1150–1154
Author information
Authors and Affiliations
Additional information
This work was partially supported by the National Natural Science Foundation of China (Grant No. 10471115)
Rights and permissions
About this article
Cite this article
Guo, Tx. Several applications of the theory of random conjugate spaces to measurability problems. SCI CHINA SER A 50, 737–747 (2007). https://doi.org/10.1007/s11425-007-0023-6
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11425-007-0023-6