It is known that the probability distribution of a continuous non-negative random variable, $X$, is uniquely determined by its associated Laplace transform,
$$L(s) = E[e^{-sX}] = \int_0^\infty e^{-sx} f(x) dx,$$
for $s ≥ 0$.
It also seems that:
(A) For any positive integer $k$,
$$L^{(k)}(s) = E[X^k e^{-sX}] = \int_0^\infty x^k e^{-sx} f(x) dx$$
uniquely determines the distribution of $X$ if $L^{(k)}(s)$ is well defined and all the lower-order moments ($E[X^{k-1}]$, $E[X^{k-2}]$, ..., $E[X]$) are known. (This is because, starting with the higher-order expression $L^{(n)}(s)$, one can solve for $L^{(n-1)}(s)$ by integration subject to the boundary condition $L^{(n-1)}(0)$ = $(-1)^{n-1}E[X^{n-1}]$. Proceeding from $n=k$ to $n=1$ ultimately yields the unique Laplace transform, $L(s) = L^{(0)}(s)$.)
My first question is whether assertion (A) is true; and if not, what is the flaw in the above argument?
A second question, assuming (A) is true, is whether the more general assertion (B) is true:
(B) For any positive real number $r$,
$$M^{(r)}(s) = E[X^r e^{-sX}] = \int_0^\infty x^r e^{-sx} f(x) dx$$
uniquely determines the distribution of $X$ if $M^{(r)}(s)$ is well defined and all the lower-order moments ($E[X^{r-1}]$, $E[X^{r-2}]$, ..., $E[X^{r-[r]}]$) are known.
Context and previous efforts:
The above questions arose out of investigation of the identifiability of $Gamma(r,\lambda)$ mixture distributions (about which I asked this earlier question: Are continuous mixtures of the gamma distribution identifiable with respect to the scale parameter?).
Further research led to the results of Boas (1939; “On a Generalization of the Stieltjes Moment Problem”, Transactions of the American Mathematical Society), which provide conditions for $E[X^r]$, $E[X^{r-1}]$, ..., $E[X^{r-[r]}]$ to characterize the distribution of $X$ uniquely.
Also, I found this helpful question and answer from a few years ago -- https://math.stackexchange.com/q/2801191 -- which addresses the case in which $L^{(k)}(1)$ is known for all non-negative integers $k$.
Thank you for reading this long post. Any guidance would be greatly appreciated!