I would suggest the classic book "The Laplace transform" by David Vernon Widder. On its page 310, Theorem 14a states the following: A necessary and sufficient condition that $f(x)$ can be expressed in the form $f(x) = \int_{0}^{\infty} e^{-xt} d \alpha(t)$, where $\alpha(t)$ is non-decreasing and bounded for $t$ in $[0,\infty)$, is that $f(x)$ should be completely monotonic for $x$ in $[0,\infty)$.

For a given $\alpha(t)$ that is non-decreasing and bounded for $t$ in $[0,\infty)$, i.e., $\alpha$ is of bounded variation on $[0,\infty)$, if $f(z) = \int_{0}^{\infty} e^{-zt} d \alpha(t)$ converges for some $z \in \mathbb{C}$, then

- there is a necessary condition on the order of $\alpha(t)$, e.g., given by Theorem 2.2a on page 39;
- Theorem 5a on page 57 gives the analyticity of $f(z)$ in a suitable region;
- if such a convergence holds absolutely on a vertical line in the complex plane $\mathbb{C}$, then Theorem 7.3 on page 66 provides the inversion formula.

Now for the two-sided integral $f(z) = \int_{-\infty}^{\infty} e^{-zt} d \alpha(t)$, just break it into the sum of $\int_{-\infty}^{0} e^{-zt} d \alpha(t)$ and $\int_{0}^{\infty} e^{-zt} d \alpha(t)$, and study both, we can obtain similar conclusions, some of which are directly stated in this book. The key to get these results is to use Dirichlet integrals, Riemann-Lebesgue lemma, and Weierstrass M-test, analytic continuation.

Finally, by a change of variable, we can study Fourier transform, Stieltjes transform, and Mellin transform of functions of bounded variation, using the same techniques. These transforms are also discussed in the book. Functions of bounded variation form a very important class of functions in probability theory, real analysis, functional analysis, and measure theory.

assumethat the analytic function $f$ can be represented by some Laplace transform, then the Bromwich inversion integral does work under reasonable conditions. $\endgroup$