If the Laplace transforms of two functions are equal, are the functions equal?

Question

In other words, is the following true? $$\int_0^\infty f(t)e^{-st}\, dt=\int_0^\infty g(t)e^{-st}\, dt\implies f(t)=g(t)$$

If not, what are examples of different functions with the same Laplace transforms (other than those that differ by a constant)?

Answer 1

No. Take

$$f(t)=0$$

and

$$g(t) = \begin{cases} 1 & t=1 \\ 0 & t\neq 1 \\ \end{cases}$$ They are different functions with the same Laplace transform.