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

Abstract: 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 $1<p<+\infty$.