Convergence in Measure under Finite Additivity

Abstract. We investigate the possibility of replacing the topology of convergence in probability with convergence in L1, upon a change of the underlying measure under finite additivity. We es-tablish conditions for the continuity of linear operators and convergence of measurable sequences, including a finitely additive analogue of Komlós Lemma. We also prove several topological impli-cations. Eventually, a characterization of continuous linear functionals on the space of measurable functions is obtained