Finitely Additive FTAP under an Atomic Reference Measure

Abstract

Let L be a linear space of real bounded random variables on the probability space $(\Omega,\mathcal{A},P_0)$. A finitely additive probability P on $\mathcal{A}$ such that

$$ P\sim P_0 \text{ and } E_P(X)=0\text{ for each }X\in L $$

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 $X_n+y\overset{a.s.}\longrightarrow Y$, for some sequence (X _n ) ⊂ L, provided EMFA’s exist and E _P (Y) = y for each EMFA P.