Positive state-price density alternative ($A_0=\mathbb{E}[\nu A_1],\nu >0$)

For deterministic initial asset values $A_0$ and a finite-dimensional terminal random vector $A_1$, exactly one possibility holds: a portfolio has nonpositive initial cost and nonnegative payoff with strict gain at at least one date, or there is a strictly positive integrable state-price density pricing $A_1$ as displayed. Apply the strictly positive barycentre cone lemma. Outside the closed support cone, strict separation gives a negative-cost portfolio. On its relative boundary, a supporting functional gives a zero-cost portfolio with positive payoff on a set of positive probability. With a traded positive bank account, normalize its discounted density to obtain an equivalent martingale measure.