= Varadhan's lemma
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{title2=$\lim_n a_n^{-1}\log\mathbb E e^{a_nf(X_n)}=\sup_x(f(x)-I(x))$}
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= Varadhan lemma
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{synonym}
= Varadhan's integral theorem
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{synonym}
If random elements satisfy a <large deviation principle> at speed $a_n\to\infty$ with <rate function> I, then a bounded continuous real function f satisfies the displayed exponential-integral identity. For the upper bound, partition its bounded range into finitely many closed intervals of width epsilon and apply the closed-set upper bound to their preimages. The largest exponential term controls the finite sum. For the lower bound, restrict to an open neighborhood where f is within epsilon of its value at a chosen point and use the open-set lower bound. Let epsilon tend to zero and optimize over the point. Boundedness avoids a separate exponential tail assumption.
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