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A positive one-period martingale deflator applied to $C=(S-B{)}^{+}$ gives $(s-b{)}^{+}\le c\le s$. If both strict payoff regions $S>B$ and $S<B$ have positive probability, then $c>0$ and $c-(s-b)>0$, proving the strict lower bound. No normalization of the positive pricing variable is needed.

Interpret positivity of $Y$ as strict positivity almost surely. Suppose the stated bounded positive pricing variable exists. For a vector $H$ satisfying the two sign restrictions,A nonnegative random variable with zero expectation vanishes almost surely. Since $Y>0$, this gives $H\cdot P=0$ almost surely and $H\cdot p=0$. The supplied nondegeneracy condition now yields $\boxed{H=0}$. This proves the direction from a one-period martingale deflator to the separating-portfolio condition; no asset-pricing theorem has been invoked.

A one-period martingale deflator is a pair $Y_0>0$, $Y_1>0$ such thatIf $Y^0,Y^1$ are deflators and ${\epsilon }_0,{\epsilon }_1>0$, their positive linear combination is strictly positive andIt is therefore another martingale deflator.