= Positive-density separation proof of the one-period asset-pricing theorem
{title2=$\text{no arbitrage}\iff\exists Q\sim P:\mathbb E_QY=0$}
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 $(1+\|Y\|)^{-1}$ 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.
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