Finitely additive equivalent martingale measures

Let $L$ be a linear space of real bounded random variables on the probability space $(Omega,mathcalA,P_0)$. There is a finitely additive probability $P$ on $mathcalA$, such that $Psim P_0$ and $E_P(X)=0$ for all $Xin L$, if and only if $c,E_Q(X)leq extess sup(-X)$, $Xin L$, for some constant $c>0$ and (countably additive) probability $Q$ on $mathcalA$ such that $Qsim P_0$. A necessary condition for such a $P$ to exist is $overlineL-L_infty^+,cap L_infty^+=$, where the closure is in the norm-topology. If $P_0$ is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability $P$ on $mathcalA$, such that $Pll P_0$ and $E_P(X)=0$ for all $Xin L$, if and only if $ extess sup(X)geq 0$ for all $Xin L$.