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An $L^{0}({\cal F},R)-$valued function's intermediate value theorem and its applications to random uniform convexity

By Guo TieXin and Zeng XiaoLin


Let $(\Omega,{\cal F},P)$ be a probability space and $L^{0}({\cal F},R)$ the algebra of equivalence classes of real-valued random variables on $(\Omega,{\cal F},P)$. When $L^{0}({\cal F},R)$ is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from $L^{0}({\cal F},R)$ to $L^{0}({\cal F},R)$. 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 $(S,\|\cdot\|)$ is random uniformly convex iff $L^{p}(S)$ is uniformly convex for each fixed positive number $p$ such that $1<p<+\infty$.Comment: 14page

Topics: Mathematics - Functional Analysis, 46A22, 46B20, 46E30
Year: 2011
DOI identifier: 10.1007/s10114-011-0367-2
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