existence of Laplace transform

Theorem 1.

For every measurable function $f\mathrm{:}\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{\infty }\mathrm{)}\mathrm{\to }\mathrm{C}$, if there exists a real number $t_{\mathrm{0}}$ such that

$${\int }_0^{\mathrm{\infty }}{e}^{-s{t}_0}|f(s)|𝑑s$$

converges, then the Laplace transform $\mathrm{L}\mathit{}\mathrm{(}f\mathrm{)}$ isa well-defined function from $\mathrm{\{}t\mathrm{\in }\mathrm{C}\mathit{}\mathrm{\mid }\mathrm{\Re }\mathit{}t\mathrm{>}\mathit{}{t}_{\mathrm{0}}\mathrm{\}}$ to$\mathrm{C}$. Furthermore, the Laplace transform function is analytic.