 Positive-density separation proof of the one-period asset-pricing theorem

Let $Y$ be the discounted gains of finitely many financial assets on any probability space. Tilting the physical probability measure by a normalized multiple of $(1+\Vert Y\Vert {)}^{-1}$ makes $Y$ integrable without changing null events. The positive-weight expectation cone alternative then says either a bounded strictly positive weight makes the weighted gain expectation zero, or a deterministic risky holding gives an arbitrage. Normalizing the weight yields an equivalent martingale measure. Conversely, under such a measure a nonnegative terminal gain with zero initial price has zero expectation, and must vanish almost surely. This proves the one-period fundamental theorem of asset pricing without assuming physical integrability of asset gains.