An L^0(F,R)-valued Function's Intermediate Value Theorem and Its Applications to Random Uniform Convexity  

作　　者：Tie Xin GUO Xiao Lin ZENG 

机构地区：[1]LMIB and School of Mathematics and Systems Science,Beihang University [2]College of Mathematics and Statistics,Chongqing Technology and Business University

出　　处：《Acta Mathematica Sinica,English Series》2012年第5期909-924,共16页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China (Grant No. 10871016)

摘　　要：Let （Ω, F, P） be a probability space and L0（F,R） the algebra of equivalence classes of real- valued random variables on （Ω, F, P）. When L0（F,R） is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0（F, R） to L0（F,R）. As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module （S, ｜｜·｜｜） is random uniformly convex iff LP（S） is uniformly convex for each fixed positive number p such that 1 〈 p 〈 ＋∞.Let （Ω, F, P） be a probability space and L0（F,R） the algebra of equivalence classes of real- valued random variables on （Ω, F, P）. When L0（F,R） is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0（F, R） to L0（F,R）. As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module （S, ｜｜·｜｜） is random uniformly convex iff LP（S） is uniformly convex for each fixed positive number p such that 1 〈 p 〈 ＋∞.

关 键 词：L0（F R）-valued function intermediate value theorem random normed module random uniform convexity modulus of random convexity 

分 类 号：O153[理学—数学]