Accumulation in the zeroes of the rate function (Large deviation principle)

Question

Consider a measure space $(\mathcal{X},\mathcal{B})$, where $\mathcal{X}$ is Polish and $\mathcal{B}$ is the Borel $\sigma$-field. Let $(\mu_n)$ be a sequence of probability measure that satisfies the Large Deviation Principle with rate function $I$, i.e. $I \geq 0$ and $I$ is lower semicontinuous. I want to show that every accumulation point of $(\mu_n)$ (every probability $\mu$ such that exist a subsequence $\mu_{n_k}$ which converges in distribution to $\mu$) is concentrated in the zeroes of $I$, that is $\mu(\{I = 0\}) = 1$.
This should be a pivotal fact for the theory but I cannot find the proof anywhere. I only showed that if $C$ is a closed set that does not contain any zeroes of $I$, $\mu_n(C) \rightarrow 0$ as $n \rightarrow \infty$. This however should not imply in general that $\mu(C)=0$, but maybe I'm missing something about the polish hypothesis or i have to add some other one.
Thanks for reading!