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Use the standard Hausdorff state-space convention, so compact sets are closed. Let $F$ be any closed set and choose $K_M$ from exponential tightness. The weak large deviation principle applies to the compact set $F\cap K_M$, whileThe principle of the largest exponential term givesLet $M\to \infty$. This is the full closed-set upper bound; the open-set lower bound was already present.

For goodness, fix $r<\infty$ and choose $M>r$. The complement $K_M^c$ is open. Its lower bound and its exponential-tightness upper estimate implyThus ${\inf }_{K_M^c}I\ge M$ and $\{I\le r\}\subset K_M$. Lower semicontinuity makes this sublevel set closed, so it is a closed subset of a compact set and therefore compact. This proves that exponential tightness upgrades a weak large deviation principle to the full principle with the same good rate function.

Work in a Hausdorff space $E$, for example a metric space. A rate function is a lower semicontinuous map $I:E\to [0,\infty ]$. A good rate function additionally has compact finite sublevel sets $\{x:I(x)\le r\}$.

A large deviation principle at large-deviation speed $L$ requires, for every open set $G$ and closed set $F$,The conventions are $\log 0=-\infty$ and $\inf \varnothing =+\infty$. A weak large deviation principle imposes the same lower bound but only the upper bound for compact sets. Exponential tightness supplies compact sets $K_M$ with $\mathrm{limsup}{L}^{-1}\log \mathbb{P}(X_L\notin K_M)<-M$ for every $M\ge 0$. The strict inequality can equivalently be obtained from the common non-strict definition by choosing a larger containment level.

Fix a finite $r\ge 0$ and choose $M>r$. By exponential tightness, choose a compact set $K_M$ with outside probability having upper exponential rate strictly below $-M$. A compact set in a Hausdorff space is closed, so its complement is an open set. The lower bound of the weak large deviation principle givesConsequently ${\inf }_{K_M^c}I>M$, and $\{I\le r\}\subset K_M$. Lower semicontinuity makes this sublevel set closed in $E$, hence a closed subset of a compact set. Thus it is compact. Every finite sublevel set is compact, so $I$ is a good rate function.

A weak large deviation principle has the usual lower bound on every open set, but requires the upper bound only on compact sets. It does not by itself control probabilities escaping every compact set. Exponential tightness provides that missing control.