Can 2 inequivalent expressions' Laplace Transforms be equal?

Question

Very simple question:

When $f(x) \neq g(x)$, is it possible for $L(f(x)) = L(g(x))$?

Answer 1

According to Lerch's theorem (see Theorem 2.1 in A. Cohen. Numerical methods for Laplace transform inversion. Springer, 2007), equality of the Laplace transforms of two functions implies that the two functions are equal almost everywhere. That said, we cannot have two functions that are substantially different from one another and have the same Laplace transform.