 Solution

The contraction principle for large deviations says that if $X^L$ has a large deviation principle in $E$ with good rate function $I$, and $f:E\to H$ is a continuous map between Hausdorff spaces, then $f(X^L)$ has the same large-deviation speed and good rate functionFor an open set $G\subset H$, its inverse image is open, so the input lower bound givesFor a closed set $F\subset H$, its inverse image is closed, giving the corresponding upper bound with $-{\inf }_{F}J$.

It remains to prove goodness, including the endpoint issue in the infimum. Each fibre $f^{-1}(\{y\})$ is closed. If $J(y)<\infty$, compact sublevel sets and lower semicontinuity make $I$ attain its infimum on that fibre: intersect it with the compact set $\{I\le J(y)+1\}$ and use the closed nested sublevels tending down to $J(y)$. Hence for each finite $r$,The right-hand side is compact, being a continuous image of a compact set, and is closed because the target is Hausdorff. Thus $J$ is lower semicontinuous and good, completing the proof.