Uniform integrability and local convexity in L0

Let L0 be the vector space of all (equivalence classes of) real-valued random variables built over a probability space (Ω,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⊆L0 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 L0-topology when restricted on convex, solid and bounded subsets of L0.