Logarithmic probabilities of open sets for a continuous rate function (${\lim }_n{a}_n^{-1}\log \mathbb{P}(X_n\in G)=-{\inf }_{G}I$)

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.