a Family of (real-valued) random variables $(X_\varepsilon)$ is said to satisfy a LDP with rate function $I$, if for any measurable set $A \in \mathcal{B}$ the following holds: \begin{align} -\inf_{x \in A^{\circ}} I(x) \leq \liminf_{\varepsilon \to 0} \varepsilon \log P(X_\varepsilon \in A) \leq \limsup_{\varepsilon \to 0} \varepsilon \log P(X_\varepsilon \in A) \leq -\inf_{x \in \overline{A}} I(x). \quad (1) \end{align} However, often times one finds statements of this sort: \begin{align} I(x) = \lim \limits_{\varepsilon \to 0} \varepsilon \log \mathbf{P}( X_\varepsilon \geq x) \quad (2) \end{align} where $x$ is greater than the mean of some limit $X_\varepsilon \to X$. Now I understand that one obtains (1) from (2) with $A=(x,\infty)$ due to the semicontinuity of $I$. However it is not at all clear to my, how one would derive (1) from (2). Is that even true in general? If not, under what conditions is it true then?
Thanks in advance,
Leo