In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.
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
基金supported by Research Foundation for Doctor Programme (Grant No. 20060055010)National Natural Science Foundation of China (Grant No. 10871101)
文摘In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.
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.
