  The Laplace transforms of positive measures on $\mathbb{R}_{+}$ converge if
and only if their distribution functions converge at continuity points of the
limiting measure. We extend this classical continuity theorem to the case of
generalised signed Radon measures. The result for the signed case requires some
additional conditions, which follow from recent results on vague convergence of
signed Radon measures. As an application, we introduce a novel Tauberian
condition for generalised signed Radon measures that extends Karamata's
Tauberian theorem.

Optimization and Control;Probability