Different proofs of uniqueness of the Laplace transform How many different types of proof do you know for the so-called Lerch's theorem, i.e., uniqueness of the Laplace transform?

I have found the following references for proofs. New books, in general, do not include the proof for Lerch's theorem.


In [1, Sections 5 and 6], there is a general proof given for Stieltjes
type Laplace transforms.
In [2, Appendix II] and [3, Section 5], the
proof uses substitution first and then approximation to a continuous
functions by polynomials.
In [4, Section 3], there is a totally
different proof, where the author makes use of the function
$E_{n}(\tau-t):=1-\mathrm{e}^{-\mathrm{e}^{nk(\tau-t)}}$ which
converges to $H(\tau-t)$ as $n\to\infty$, where $H$ is Heaviside's
unit function.


Do you know other proofs for Lerch's theorem.
Can you redirect me to some other proofs?

References
[1] D. V. Widder, The Laplace Transform, 
Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941.
[2] D. L. Kreider, R. G. Kuller, D. R. Ostberg, F. W. Perkins, An Introduction to Linear Analysis, Addison-Wesley Publishing Co., Inc., Reading, Mass.-Don Mills, Ont. 1966.
[3] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, New York-Heidelberg, 1974.
[4] E. J. Watson, Laplace Transforms and Applications, VNR New
    Mathematics Library, 10. Van Nostrand Reinhold Co., New York-London,
    1981.
I don't think this question has a well defined answer.  As far as I know, all proofs of the uniqueness of the Laplace transform are essentially corollaries of this one statement:

$\displaystyle \int_{0}^{\infty} f(t) \text{e}^{-st} dt = 0 \implies f(t) = 0$

The number of statements equivalent to this is infinite (e.g. you could add $1$ to both sides of the equation, to get a stupid yet true statement).  Perhaps you mean to restrict the proofs to those which are reasonably simple and useful, but then we start to get into highly subjective territory.
I don't think this question has a well defined answer.  As far as I know, all proofs of the uniqueness of the Laplace transform are essentially corollaries of this one statement:

$\displaystyle \int_{0}^{\infty} f(t) \text{e}^{-st} dt = 0 \implies f(t) = 0$

The number of statements equivalent to this is infinite (e.g. you could add $1$ to both sides of the equation, to get a stupid yet true statement).  Perhaps you mean to restrict the proofs to those which are reasonably simple and useful, but then we start to get into highly subjective territory.