Finitely additive equivalent martingale measures

Let $L$ be a linear space of real bounded random variables on the probability space $(Ω,\mathcal{A},P_0)$. There is a finitely additive probability $P$ on $\mathcal{A}$, such that $P\sim P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $c\,E_Q(X)\leq\text{ess sup}(-X)$, $X\in L$, for some constant $c>0$ and (countably additive) probability $Q$ on $\mathcal{A}$ such that $Q\sim P_0$. A necessary condition for such a $P$ to exist is $\bar{L-L_\infty^+}\,\cap L_\infty^+=\{0\}$, 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 $\mathcal{A}$, such that $P\ll P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $\text{ess sup}(X)\geq 0$ for all $X\in L$.

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