Good rate function ($I$)

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.

If a good rate function has unique zero mu, every closed set missing mu has strictly positive infimum rate, possibly infinity. Otherwise a sequence of rates tending to zero lies eventually in the compact unit sublevel set; a subsequence converges to a point of the closed set, and lower semicontinuity makes that point another zero. Compactness also ensures existence of a zero whenever the infimum is zero.