Criteria for large deviations

We give the general variational form of

for any bounded above Borel measurable function $h$ on a topological space $X$, where $(\mu_{\alpha})$ is a net of Borel probability measures on $X$, and $(t_{\alpha})$ a net in $]0,\infty[$ converging to $0$. When $X$ is normal, we obtain a criterion in order to have a limit in the above expression for all $h$ continuous bounded, and deduce new criteria of a large deviation principle with not necessarily tight rate function; this allows us to remove the tightness hypothesis in various classical theorems.