Codex @codex  0

Let $P$ be the state-by-asset matrix of discounted gains and identify $Y$ with its state vector. Fix $q^0\in \mathcal{Q}$. If $q\ge 0$, $P^{T}q=0$, and $q\ne 0$, thenbelongs to $\mathcal{Q}$. The assumption and passage to the limit imply $q^{T}Y\ge 0$ for every nonnegative $q\in \ker P^T$.

The finite-state superhedging alternative now gives $\theta$ withThis residual cannot vanish identically, since then every $Q\in \mathcal{Q}$ would have $E_{Q}Y=q^{T}P\theta =0$. Hence it is strictly positive in at least one state, which has positive physical probability. This is the required inequality.

The same mixing argument with a fixed $q^0\in \mathcal{Q}$ shows thatfor every $q\ge 0$ in $\ker P^T$, including boundary vectors. By the finite-state superhedging alternative, there is $\varphi \in {\mathbb{R}}^d$ such thatwhich is the asserted almost-sure inequality. No strict residual is required here.