I am reading the first few pages of Dembo&Zeitouni. I have some questions about the alternative definition of the large deviation principle in page 6. The equation (1.2.7) states that
For every $\alpha < \infty$ and every measurable set $\Gamma$ with $\bar{\Gamma} \subset \Psi_I(\alpha)^c$, $$\limsup_{\epsilon \to 0} \epsilon \log\mu_{\epsilon} (\Gamma) \leq -\alpha.$$
is equivalent to the definition to the LDP upper bound $$\limsup_{\epsilon \to 0} \epsilon \log\mu_{\epsilon} (\Gamma) \leq -\inf_{x\in \bar{\Gamma}} I(x) $$.
It is not clear to me that why they are equivalent.One direction is obvious. However, why the former implies the latter is not clear to me. I tried to take $\alpha = \inf_{x\in \bar{\Gamma}} I(x)$, however, I can not show $\bar{\Gamma}$ is a subset of $\Psi_I(\alpha)^c$.
Can anyone shed some light on this? Thank you in advance.