Criteria for large deviations

We give the general variational form of limsup(integral(X) e(h(x)/t alpha) mu(alpha)(dx))(t alpha) 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,infinity[ 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.