Exponential decay of measures and Tauberian theorems

We study behavior of a measure on $[0,\infty)$ by considering its Laplace transform. If it is possible to extend the Laplace transform to a complex half-plane containing the imaginary axis, then the exponential decay of the tail of the measure occurs and under certain assumptions we show that the rate of the decay is given by the so called abscissa of convergence and extend the result of Nakagawa from [Nak05]. Under stronger assumptions we give behavior of density of the measure by considering its Laplace transform. In situations when there is no exponential decay we study occurrence of heavy tails and give an application in the theory of non-local equations.