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We consider a general set $\mathcal{X}$ of adapted nonnegative stochasticprocesses in infinite continuous time. $\mathcal{X}$ is assumed to satisfy mildconvexity conditions, but in contrast to earlier papers need not contain astrictly positive process. We introduce two boundedness conditions on$\mathcal{X}$ -- DSV corresponds to an asymptotic $L^0$-boundedness at thefirst 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 theexistence of a strictly positive adapted process $Y$ such that $XY$ is asupermartingale for all $X \in \mathcal{X}$, with an additional asymptoticstrict positivity property for $Y$ in the case of DSV.

Characterisation of $L^0$-boundedness for a general set of processes with no strictly positive element