Let $\mathcal{K}_s$ be $$ \mathcal{K}_s=\{\tilde{V}_t(\theta):0\leq t<\infty,\,\theta\text{ a simple strategy}\},$$ where $\tilde{V}_t(\theta)$ is the discounted value process of the self financing strategy $\theta$, $$ \tilde{V}_t(\theta)=x+\sum_{j=1}^k\sum_{i=0}^{m-1} a^j_{t_i}(\tilde{S}^j_{t_{i+1}\wedge t}-\tilde{S}^j_{t_i\wedge t}).$$
Now let $\mathcal{U}$ be $$ \mathcal{U}=\{f-h:f\in\mathcal{K}_s,\;h\in L_+^\infty\},$$ where $$L_+^\infty(P) = \{ Z\in L^\infty(P):P(Z\geq0)=1\}. $$
Suppose there exists $g \in \mathcal L^q$ with $P(g>0)=1$ such that $$\int fgdP\leq0\hspace{1cm}\forall f\in\mathcal{U}.$$
I want to show that this implies that the discounted stock prices $(\tilde S_t^1,\tilde S_t^2,....,\tilde S_t^k)$ are $Q$-martingales where $dQ=gdP$.
My attempt: I know that if $\int fg dP=0$ for all $f$ in $\mathcal K_s$ then $(\tilde S_t^1,\tilde S_t^2,....,\tilde S_t^k)$ are $Q$-martingales because $1_A (\tilde S_t^i-\tilde S_s^i) \in \mathcal {K}_s$ for $A\in \mathcal G_s$ and $0 \leq s \leq t$. Can anyone please help me with the above case.