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

Abstract

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

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