  We consider a general set $\mathcal{X}$ of adapted nonnegative stochastic
processes in infinite continuous time. $\mathcal{X}$ is assumed to satisfy mild
convexity conditions, but in contrast to earlier papers need not contain a
strictly positive process. We introduce two boundedness conditions on
$\mathcal{X}$ -- DSV corresponds to an asymptotic $L^0$-boundedness at the
first time all processes in $\mathcal{X}$ vanish, whereas NUPBR$_{\rm loc}$
states that $\mathcal{X}_t = \{ X_t : X \in \mathcal{X}\}$ is bounded in $L^0$
for each $t \in [0,\infty)$. We show that both conditions are equivalent to the
existence of a strictly positive adapted process $Y$ such that $XY$ is a
supermartingale for all $X \in \mathcal{X}$, with an additional asymptotic
strict positivity property for $Y$ in the case of DSV.
