The Upper Semicontinuous Envelope of a Supremum of Plan-Jet Viscosity Subsolutions is a Plan-Jet Viscosity Subsolution

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\rho>0$ be real, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ be uniformly continuous on bounded sets, and let $(\mathrm{E})$ be the discounted stationary Hamilton--Jacobi equation with discount rate $\rho$ and Hamiltonian $\mathcal{H}$; metrics are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics. Let $\mathcal{F}$ be a nonempty set of upper semicontinuous functions $\Sigma^{2}_{d}\to\mathbb{R}$, each a plan-jet viscosity subsolution of $(\mathrm{E})$, and let $K\in\mathbb{R}$ satisfy $v(\nu)\le K$ for all $v\in\mathcal{F}$ and $\nu\in\Sigma^{2}_{d}$. Let $w:\Sigma^{2}_{d}\to\mathbb{R}$, $w(\nu)=\sup\{v(\nu):v\in\mathcal{F}\}$, the least upper bound of a nonempty set of reals bounded above by $K$, as recorded in The Real Numbers: Standing Notation and Background §bounds, so that $w\le K$; and let $w^{*}$ be its upper semicontinuous envelope, which exists because $w$ is bounded above.

¶Then $w^{*}$ is a plan-jet viscosity subsolution of $(\mathrm{E})$.