Varadhan’s lemma for large deviations

If K is a nonempty compact subset of X and f : X → [−∞,∞] is lower semicontinuous, there is some x0 ∈ K such that f(x0) = infx∈K f(x). A rate function is a function I : X → [0,∞] such that (i) there is some x ∈ X with I(x) <∞ and (ii) for any c ∈ [0,∞], f−1([0, c]) is compact (namely, the sublevel sets of I are compact). Condition (ii) implies that a rate function is lower semicontinuous. For S ∈ BX , the Borel σ-algebra of X , we define