The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure
The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure
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DOI：
10.1007/s11425-008-0047-6
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发表时间：
2008-08
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影响因子：
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通讯作者：
T. Guo
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作者：
T. Guo
设（Ω，A，μ）是概率空间，K是真实的数的标量场R或复数的标量场C，（S，X）是K上以（ω，A，μ）为基的随机赋范空间.用E表示（S，X）的支集，即E是集合{A∈A：存在S中的元素使得对几乎所有的ω，Xp（ω）> 0}的本质上确界.本文首先建立了随机赋范空间中的Banach-Alaoglu定理：（S，X）的随机共轭空间（S*，X*）的随机闭单位球S *（1）= {f∈S*：X*f <$1}在（S*，X *）上的随机弱星星拓扑下是紧的，iffE <$A=：{E <$A| A∈A}本质上是纯μ-原子的（即存在一个不相交的族{An：n∈N}，它由至多可数个μ-原子组成，使得E = F Δ H =1∞ A n，并且对于每个元素FinE Δ A，在由{An：n∈N}生成的σ-代数中存在一个H，满足μ（FΔH）= 0），它的证明迫使我们提供一个关键的拓扑技巧，因此比相应的经典情况复杂得多。进一步，建立了完备随机赋范模中的Banach-Bourbaki-Kakutani-Šmulian（简称BBKS）定理：若（S，X）是完备随机赋范模，则（S，X）的随机闭单位球S（1）= {p∈S：Xp <$1}在（S，X）上的随机弱拓扑下是紧的当且仅当（S，X）是随机自反的且E <$A本质上是纯μ-原子的.我们最近的工作证明了著名的经典James定理对于任意完备随机赋范模仍然成立，即完备随机赋范模是随机自反的当且仅当其上任意几乎处处有界随机线性泛函的随机范数在其随机闭单位球上可达，但本文表明，经典的巴拿赫-Alaoglu定理和BBKS定理对于完备随机赋范模不普遍成立，除非它们具有极其简单的层结构，即它们的载体基本上是纯μ原子的。在完全随机赋范模中结合James定理和BBKS定理直接导致一个有趣的现象：存在许多著名的经典命题，它们在Banach空间中是相互等价的，其中一些在任意完备随机赋范模的上下文中仍然是相互等价的，而另一个在任意完备随机赋范模的上下文中不再等价于另一个，除非所讨论的随机赋范模具有极其简单的分层结构这种现象是在随机度规理论的发展过程中首次发现的。
Let (Ω,A,μ) be a probability space,Kthe scalar fieldRof real numbers orCof complex numbers,and (S,X) a random normed space overKwith base (ω,A,μ). Denote the support of (S,X) byE, namelyEis the essential supremum of the set {A∈A: there exists an elementpinSsuch thatXp(ω) > 0 for almost all ω inA}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ballS*(1) = {f∈S*:X*f⩽ 1} of the random conjugate space (S*,X*) of (S,X) is compact under the random weak star topology on (S*,X*) iffE∩A=: {E∩A|A∈A} is essentially purely μ-atomic (namely, there exists a disjoint family {An:n∈N} of at most countably many μ-atoms fromE∩Asuch thatE= ∪n=1∞Anand for each elementFinE∩A, there is anHin the σ-algebra generated by {An:n∈N} satisfying μ(FΔH) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S,X) is a complete random normed module, then the random closed unit ballS(1) = {p∈S:Xp⩽ 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive andE∩Ais essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of random metric theory.