The Structure Condition and Uniform Continuity on Bounded Sets Imply Their Forms at Bounded Positions

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$.

1. (Structure condition)¶ If $\mathcal{H}$ satisfies the structure condition, then it satisfies the structure condition at bounded positions.

2. (Momentum continuity)¶ If $\mathcal{H}$ is uniformly continuous on bounded sets, then it is uniformly continuous in the momentum at bounded positions.