= Solution
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> $\{x:I(x)\leq r\}$.
A <large deviation principle> at <large-deviation speed> $L$ requires, for every <open set> $G$ and <closed set> $F$,
$$
-\inf_{x\in G}I(x)\leq\liminf_{L\to\infty}\frac1L\log\mathbb P(X_L\in G),\qquad
\limsup_{L\to\infty}\frac1L\log\mathbb P(X_L\in F)\leq-\inf_{x\in F}I(x).
$$
The conventions are $\log0=-\infty$ 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 $\limsup L^{-1}\log\mathbb P(X_L\notin K_M)<-M$ for every $M\geq0$. The strict inequality can equivalently be obtained from the common non-strict definition by choosing a larger containment level.
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