It is true that the Laplace transform of a (positive) random variable characterises that random variable, just like its density?
($L_X(z) = E(exp(-Xz))$)
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Up to a change of sign in the exponent, the Laplace transform is the moment generating function.
The MGF only characterizes a random variable if the MGF converges in an open interval around 0.
This isn't automatically the case, even when all moments exist (e.g. see the lognormal).
By contrast the characteristic function (which up to sign in the exponent is the same as the Fourier transform) does characterize a random variable.
For more information, see this question, and the several threads linked to from there.