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

Abstract

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