Variational form of the large deviation functional

Under exponential tightness hypothesis, we show that the large deviation functional Λ¯(·) has the same variational form as when large deviations hold. As an application, we identify the affine regularization of a rate function on any real Hausdorff topological vector space in terms of Λ¯(·). Some classical results involving the generalized log-moment generating function are improved.