I have the following problem. Assume that $u$ is locally of bounded variation function on $(0,\infty)$ and can be of the form $u(y)=\bar{\mu}(y)-e^{-y}, y>0,$ where $\bar{\mu}(y)=\int_{y}^\infty\mu(du)$ with $\int_{0}^\infty\min\{u,1\}\mu(du)<\infty$. We know that $$h(z)=\int_{0}^{\infty}e^{-zy}u(y)dy$$ is absolutely convergent for $\Re(z)>-1$ and $h$ can be extended to an entire function. Does this imply that $e^{-zy}u(y)$ is absolutely integrable on $[0,\infty)$ for every complex number z?
I would appreciate any hint and perhaps some reference to a body of literature on Laplace transforms of signed measure provided you are aware of such.
Thanks in advance!