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Uniform integrability and local convexity in $L^0$

Let $L^0$ be the vector space of all (equivalence classes of) real-valued random variables built over a probability space $(Ω, \mathcal{F}, P)$, 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 $L^0$.