  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 $L^0$.
