a good rate function

A good rate function is a rate function with one extra and very useful property: its sublevel sets are compact, not merely closed. This is the distinction between an LDP and a 'good' LDP, and it is the compactness condition that makes the whole machinery of large deviations behave like a clean variational calculus rather than a delicate exercise in topology.

Concretely, I: X -> [0, infinity] is a good rate function if every set {x : I(x) <= c} is compact for each finite c. Two consequences follow immediately and are why one wants goodness. First, infima of I over closed sets are actually attained — there exists a minimiser, the 'most likely way' for a rare event to happen, so statements like 'the rare event happens by following the cheapest path' are literally true rather than approximate. Second, goodness is exactly the hypothesis under which the contraction principle and Varadhan's lemma can be applied; pushing a good LDP through a continuous map gives a good LDP, and the resulting inf-convolution is well posed.

Goodness is also the natural partner of exponential tightness. A sequence is exponentially tight if for every L there is a compact K with the complement having probability decaying faster than e^(-L n); exponential tightness plus an LDP-style upper bound on compacts upgrades to a full LDP with a good rate function. In Polish spaces this is how most LDPs are actually proven. The classic Cramer rate function in R^d is good whenever the log-moment generating function is finite in a neighbourhood of the origin (steepness); without that, I can have non-compact level sets and goodness can fail.