Perron's Method on Unitary Laws: the Upper Envelope of a Bounded Family of Subsolutions Is a Subsolution

In the setting of Unitary Laws with Free Unitary Noise: Standing Data, with the conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces, let $H$ be an admissible Hamiltonian, let $\mathcal{S}$ be a nonempty set of viscosity subsolutions of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H$, and suppose there is a real $M$ with $u(\lambda)\le M$ for every $u\in\mathcal{S}$ and every $\lambda\in\mathcal{L}_{d}$. For $\lambda\in\mathcal{L}_{d}$ let $w(\lambda)=\sup\{u(\lambda):u\in\mathcal{S}\}$, which exists by The Real Numbers: Standing Notation and Background §bounds, the set being nonempty and bounded above by $M$.

Then the following hold.

1. (Boundedness)¶ The map $w:\mathcal{L}_{d}\to\mathbb{R}$ is bounded.

2. (Upper envelope)¶ The upper semicontinuous envelope $w^{*}$ of $w$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$ is a viscosity subsolution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H$.