= Logarithmic probabilities of open sets for a continuous rate function
{title2=$\lim_na_n^{-1}\log\mathbb P(X_n\in G)=-\inf_GI$}
If a <large deviation principle> has a finite <continuous> <rate function>, then the infima on any <open set> and its <closure> agree. Approximate every closure point by points of the set and use continuity. The open-set lower and closed-set upper bounds consequently match, giving the displayed logarithmic limit, including the empty-set convention. A single-point event need not obey such an equality.
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