Convergence in measure under finite additivity

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 establish conditions for the continuity of linear operators and convergence of measurable sequences, including a finitely additive analog of Komlós Lemma. We also prove several topological implications. Eventually, a characterization of continuous linear functionals on the space of measurable functions is obtained