Title: Uniform integrability and local convexity in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">L</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msup></mml:math>
Abstract: 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.