  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.
