= Positive state-price density alternative
{title2=$A_0=\mathbb E[\nu A_1],\quad\nu>0$}
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>.
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