 Good rate function

A rate function is a lower semicontinuous map $I:E\to [0,\infty ]$. It is good if every sublevel set $\{I\le M\}$ is a compact set. This property controls escapes to infinity and makes the minimizations in the contraction principle for large deviations attain their finite infima. On ${\mathbb{R}}^d$, a lower semicontinuous rate function tending to infinity with the norm is good.