Is the inverse of the Laplace transform unique?

[Bipolarity]

I've been wondering whether the Laplace transform is injective. Suppose I have that
\int^{∞}_{0}e^{-st}f(t)dt = \int^{∞}_{0}e^{-st}g(t)dt for all s for which both integrals converge. Then is it true that f(t) = g(t) ? If so, any hints on how I might prove it?

Thanks!

BiP

 

[HallsofIvy]

That would mean that
\int_0^\infty e^{-st}(f(t)- g(t))dt= 0

Does that necessarily mean that f(t)= g(t)?