Finitely additive FTAP under an atomic reference measure

Abstract

Let L be a linear space of real bounded random variables on the probability space (Ω, A, P 0). A finitely additive probability P on A such that P ∼ P 0 and E P(X) = 0 for each X ∈ L is called EMFA (equivalent martingale finitely additive probability). In this note, EMFA's are investigated in case P 0 is atomic. Existence of EMFA's is characterized and various examples are given. Given y ∈ ℝ and a bounded random variable Y, it is also shown that X n + y a.s. → Y, for some sequence (X n) ⊂ L, provided EMFA's exist and E P (Y) = y for each EMFA P. © 2012 Springer-Verlag.