I am currently writing a thesis that serves as an introduction to the theory of large deviations for people on about my level. The task is to closely follow Varadhan's booklet "Large Deviations and Applications". I have some trouble understanding the proof of Theorem 2.4, which is a modification of the contraction principle. In particular, I have trouble understanding what Varadhan did not prove in the book. Theorem 2.4 states the following:
Let $(\mathbb{P}_n)_{n\in\mathbb{N}}$ satisfy the large deviation principle with a rate function $I(\cdot)$ on a complete separable metric space $\Omega$. Let $(F_n)_{n\in\mathbb{N}}$ be a sequence of continuous maps $F_n:\Omega\rightarrow\Omega'$, where $\Omega'$ is another complete separable metric space. Assume that $\lim_{n\rightarrow\infty}F_n=F$ exists uniformly over compact subsets of $X$. Then if we define $(\mathbb{Q}_n)_{n\in\mathbb{N}}$ on $\Omega'$ by $\mathbb{Q}_n=\mathbb{P}_n\circ F_n^{-1}$, then $(\mathbb{Q_n})_{n\in\mathbb{N}}$ satisfies the large deviation principle with a rate function $J(\cdot)$ defined by $$J(y):=\inf\limits_{x\in\Omega:\ F(x)=y}I(x).$$
I understand that to prove this, we have to check all 5 properties of the definition in his book. I struggle with the second and third property, as they seemed to be too trivial for Varadhan to explain. So we have to prove
(ii) $J(\cdot)$ is lower semi-continuous.
(iii) For each $k<\infty$ the set $\{x\in\Omega'|J(x)\leq k\}$ is compact.
Note that these two properties already hold for the function $I(\cdot)$.
I do have some trouble understanding these basic concepts and as to how to prove these properties in a correct manner. I have not really worked with semi-continuity before and am therefore unsure what definition would be appropriate to use here, since we can use the one for general topologic spaces as well as the one for metric spaces.
Can someone help me out a bit so I can develop a better understanding of this particular theorem? I also wonder what the exact meaning of "limit exists uniformly over compact subsets" is.
Uniformly over compact subsets means exactly what it says: for every compact set $K\subseteq\Omega$, $F_n\to F$ uniformly on $K$. In particular, this implies $F$ is also continuous.
Note that lower semicontinuity is equivalent to $\{y:J(y)\le k\}$ is closed for every $k$, so it is sufficient to prove (iii). To show that this set is compact, let $\{y_n\}$ be a sequence in $\Omega'$ with $J(y_n)\le k$ for all $n$. There exists $\{x_n\}\subset\Omega$ with $F(x_n)=y_n$ and $I(x_n)\le k+\frac1n$. In particular, $x_n\in\{z:I(z)\le k+1\}$ for all $n$, so since $I$ has compact level sets we know there is a subsequence $\{n_j\}$ such that $x=\lim_{j\to\infty}x_{n_j}$ exists. Note that, for all $\ell\ge j$, we have $I(x_{n_\ell})\le k+\frac1{n_\ell}\le k+\frac1{n_k}$, so since $I$ is lower semicontinuous we find that $I(x)\le k+\frac1{n_j}$ by letting $\ell\to\infty$. Now letting $j\to\infty$ we see that $I(x)\le k$. Let $y:=F(x)$. We know $J(y)\le I(x)\le k$, and by continuity of $F$ we know $y_{n_j}\to y$. Hence $\{y:J(y)\le k\}$ is compact.