Price uniqueness and FTAP with finitely additive probabilities

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 egin{equation*} Psim P_0quad ext{and}quad E_P(X)=0 ext{ for each }Xin L end{equation*} is called EMFA (equivalent martingale finitely additive probability). In this paper, EMFA's are investigated in case $P_0$ is atomic. Existence of EMFA's is characterized and a question raised in cite{BPR} is answered. Some results of the following type are obtained as well. Let $yinmathbb{R}$ and $Y$ a bounded random variable. Then $X_n+yoverset{a.s.}longrightarrow Y$, for some sequence $(X_n)subset L$, provided EMFA's exist and $E_P(Y)=y$ for each EMFA $P$.