Finitely Additive FTAP under an Atomic Reference Measure

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