I have a question regarding the inverse Laplace transform (LT).
I have searched for the formal definition and properties of inverse LT and I know that if $$ \mathcal{L}\{f(t)\}(s) = \mathcal{L}\{g(t)\}(s) \quad \forall s>a, $$ where $f, g$(defined on $[0,\infty)$) are exponential type $a$, then $f(t) = g(t)~\forall t\ge 0$.
My question is: for a general Laplace transform, if $F(s) = \mathcal{L}\{f(t)\}(s)$ exist for $s\in (a, \infty)$, and $\mathcal{L}\{g(t)\}(s) = F(s)$ for $s \in (b, \infty), b>a$. Can we draw the conclusion that $g(t)$ is the inverse Laplace transform of $F(s)$?