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 ...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 〈 ＋∞.展开更多
基金Supported by National Natural Science Foundation of China (Grant No. 10871016)
文摘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 〈 ＋∞.
