If $\mathcal A:=\{X\in \mathcal X:\rho(X)\le0\}$ with $\rho:\mathcal X(\Omega,\mathcal F)\to\mathbb R$ with $\mathcal X$ being the linear space of bounded functions containing constants, then $$\inf\{m\in\mathbb R:m\in \mathcal A\}>-\infty$$
$\rho$ has some other properties, but I don't know if they're necessary. $\mathcal A$ is closed in $L^{\infty}$ w.r.to $||\cdot||_{\infty}$ norm. So if the infimum is $-\infty$ then since $\mathcal A$ is closed it cannot be in $L^{\infty}$, is it something like this ?
(This is Proposition $4.6$ from the book '' Stochastic Finance'' by Hans Föllmer & Alexander Schied)
EDIT: here is the detailed definition of $\rho$