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 ...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.展开更多
The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a modu...The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a module over the algebra L(μ, K) frequently makes our investigations into random duality theory considerably different from the corresponding ones into classical duality theory, thus in this paper we have to first begin in overcoming several substantial obstacles to the study of stratification structure on random locally convex modules. Then, we give the representation theorem of weakly continuous canonical module homomorphisms, the theorem of existence of random Mackey structure, and the random bipolar theorem with respect to a regular random duality pair together with some important random compatible invariants.展开更多
基金the National Natural Science Foundation of China (Grant No. 10471115)
文摘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.
基金supported by National Natural Science Foundation of China (Grant No. 10871016)
文摘The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a module over the algebra L(μ, K) frequently makes our investigations into random duality theory considerably different from the corresponding ones into classical duality theory, thus in this paper we have to first begin in overcoming several substantial obstacles to the study of stratification structure on random locally convex modules. Then, we give the representation theorem of weakly continuous canonical module homomorphisms, the theorem of existence of random Mackey structure, and the random bipolar theorem with respect to a regular random duality pair together with some important random compatible invariants.
