= Exponential tightness upgrades a weak large deviation principle
Suppose a <weak large deviation principle> with <rate function> $I$ holds on a <Hausdorff space>, at speed $a_n\to\infty$, and the random elements are <exponentially tight>. For a finite $r$, choose a <compact set> $K$ whose complement has upper exponential rate strictly below $-M$, with $M>r$. The open-set lower bound on $K^c$ implies $\inf_{K^c}I>M$, so $\{I\leq r\}\subset K$. <Lower semicontinuity> makes this <sublevel set> closed, hence compact. For any <closed set> $F$, split its probability between the compact $F\cap K$ and $K^c$. The upper exponential rate is at most $\max\{-\inf_F I,-M\}$. Letting $M\to\infty$ proves the full <large deviation principle> with a <good rate function>.
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