Uniform integrability and local convexity in $L^0$

TL;DR

This paper characterizes when a set of random variables in $L^0$ can be made uniformly integrable under an equivalent measure and links this to local convexity of the topology on certain subsets.

Contribution

It provides a necessary and sufficient condition for uniform integrability under an equivalent measure and relates this to local convexity in the $L^0$ topology.

Findings

Characterizes sets in $L^0$ that are uniformly integrable under some equivalent measure.

Establishes a connection between uniform integrability and local convexity of the $L^0$ topology.

Provides a measure-free perspective on uniform integrability in $L^0$.

Abstract

Let $L^0$ be the vector space of all (equivalence classes of) real-valued random variables built over a probability space $(\Omega, \mathcal{F}, P)$, equipped with a metric topology compatible with convergence in probability. In this work, we provide a necessary and sufficient structural condition that a set $X \subseteq L^0$ should satisfy in order to infer the existence of a probability $Q$ that is equivalent to $P$ and such that $X$ is uniformly $Q$-integrable. Furthermore, we connect the previous essentially measure-free version of uniform integrability with local convexity of the $L^0$-topology when restricted on convex, solid and bounded subsets of $L^0$.