I've often heard the claim that the Laplace transform reduces functions to sums (or rather integrals) of a bunch of exponential functions, like a Fourier series reduces things to sums of sinusoids and a Fourier transform reduces things to integrals of sinusoids.
While this is clear for the Fourier transform, which is its own inverse, I'm having trouble understanding how this is so for the Laplace transform.
The Laplace transform of $e^{at}$, for example, is $1/(s-a)$. If the "integral of exponentials" thing was correct, it would've been $\delta(s-a)$. The integral $\int_0^\infty e^{st}/(s-a) ds$ actually diverges.
It seems to me that it is not $f(t)$, but rather $F(s)$, which is written as an integral of exponentials of the form $e^{-st}$, and the weight on each $t$ is given by $f(t)$. Am I right?