Definition 16.11.3A4F165851008

Definition 16.11.3.A4F165851008 Let $f\colon X\to [-\infty ,+\infty ]$ be a Borel function, called an obstacle, and put

The members of $\mathcal A_\theta (f)$ are called the competitors for $f$. If $\mathcal A_\theta (f)$ is nonempty and uniformly bounded above, the obstacle envelope of $f$ is

where ${}^*$ denotes the upper semicontinuous regularization. If $\mathcal A_\theta (f)$ is empty, we put $P_\theta (f)=-\infty $. If $\mathcal A_\theta (f)$ is nonempty but not uniformly bounded above, $P_\theta (f)$ is not defined, and whenever we use $P_\theta (f)$ for an obstacle that is not bounded above, we verify the upper bound first. If $f\leq b$ quasi-everywhere for a constant $b$, the upper bound is automatic: every competitor satisfies $v\leq b$ quasi-everywhere, hence almost everywhere, and therefore everywhere by lemma 6.14.4(1).