Boundedness and Uniform Continuity of Population Cost Data on a Compact Control Set

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, let $(L,G)$ be population cost data on $l$ states with control dimension $m$, let $\Delta^l$ be the probability simplex, and let $\mathcal{A}$ be a nonempty subset of $\mathbb{R}^m$ that is compact for the topology determined by the Euclidean distance.

1. (Boundedness.) There is a finite real number $C\ge0$ such that $|L(\Sigma,\alpha)|\le C$ and $|G(\Sigma)|\le C$ for every $\Sigma\in\Delta^l$ and every $\alpha\in\mathcal{A}$.

2. (Uniform continuity in the state, uniformly in the control.) For every real $\varepsilon>0$ there is a real $\delta>0$ such that

$$|L(\Sigma,\alpha)-L(\Sigma',\alpha)|\le\varepsilon\qquad\text{and}\qquad|G(\Sigma)-G(\Sigma')|\le\varepsilon$$

whenever $\Sigma,\Sigma'\in\Delta^l$ satisfy $|\Sigma-\Sigma'|\le\delta$ and $\alpha\in\mathcal{A}$.