As in many classic textbooks, the definitions of Large Deviation Principle is as follows:
$\{\mu_n\}$ has LDP with speed $a_n$ and rate $I(x)$ if the following holds for any measurable $A$: $$\limsup\frac{1}{a_n}\log\mu_n(A)\leq -\inf_{x\in\bar{A}}I(x),$$ $$\liminf\frac{1}{a_n}\log\mu_n(A)\geq -\inf_{x\in A^{o}}I(x).$$
I was wondering why we need the closure and interior as upper and lower bounds here(resp.), instead of use the infimum on $A$ directly.