Laplace transform of a signed measure

Question

I'm interested in the Laplace transform $\mathcal{L}\{\mu\}(z) = \int_{[0,\infty)}e^{-zt}\mu(\text{d}t)$ of signed measure $\mu$ on $[0,\infty)$.

1. Is there a body of literature on this topic? Can anyone recommend a comprehensive reference?
2. Are there some natural conditions on $\mu$ that are equivalent to the condition that $\mathcal{L}\{\mu\}(z) > 0$ for all $z > 0$?
3. If 2. cannot be answered in the general setting, how about the case were $\mu$ is a finite, weighted sum of Dirac measures?

Answer 1

The typical kind of condition for positivity typically involves positivity of matrices. For example, $$ \sum_{m=1}^{N}\sum_{n=1}^{N}a_n\overline{a_m}\mathcal{L}\{\mu\}(z_n+\overline{z_m}) \\ = \sum_{m=1}^{N}\sum_{n=1}^{N}a_n\overline{a_m}\int_{0}^{\infty}e^{-tz_n}e^{-t\overline{z_m}}d\mu(t) \\ = \int_{0}^{\infty}\sum_{m}\sum_{n}a_ne^{-tz_n}\overline{a_me^{-tz_m}}d\mu(t) \\ = \int_{0}^{\infty}\left|\sum_{n=1}^{N}a_n e^{-tz_n}\right|^2d\mu(t) \ge 0. $$ So, if $\mu$ is a positive measure, then all of the sums above are non-negative. Positivity should be equivalent to $\{ \mathcal{L}\{\mu\}(z_n+\overline{z_m})\}_{n,m=1}^{N}$ being a positive definite matrix for all finite sequences $\{ z_n \}_{n=1}^{N}$ in the right half plane.