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

TL;DR

This paper establishes an intermediate value theorem for continuous local functions on the space of real-valued random variables and applies it to characterize random uniform convexity in random normed modules.

Contribution

It proves a new intermediate value theorem for $L^{0}({cal},R)$-valued functions and links random uniform convexity of modules to classical uniform convexity of associated $L^{p}$ spaces.

Findings

Proved an intermediate value theorem in the topology of convergence in probability.

Derived expressions for the modulus of random convexity.

Characterized random uniform convexity via $L^{p}$ spaces.

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 $1<p<+\infty$.