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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. 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