Definition:Good Rate Function

Definition
Let $\XX$ be a topological space.

Let $\overline \R_\ge$ be the positive extended real number line.

Let $I : \XX \to \overline \R_\ge$ be a rate function.

Then $I$ is good for all $\alpha \in \R_{\ge 0}$:
 * $\operatorname {lev} \limits_{\mathop \le \alpha} I$ is compact

where $\operatorname {lev} \limits_{\mathop \le \alpha} I$ denotes the lower level set of $I$ at $\alpha$.