On the existence of a certain real measure I'm stuck on the following problem :
Does there exist a real borel measure $\mu$ on $[0,1]$ such that
$$\int_{0}^{1}x^n d\mu = e^{-n^2}$$
for all $n \geq 1$ ?
Does anyone have any hint?
Thank you
 A: If $\mu$ is supposed to be positive, then $\langle f,g\rangle=\int fgd\mu$ defines a pre-inner product on the real vector space of polynomial functions on $[0,1]$.  Gramian matrices with respect to this inner product must be positive semidefinite.  For example, 
$\begin{bmatrix}
\langle x,x\rangle & \langle x,x^2\rangle \\
\langle x^2,x\rangle & \langle x^2,x^2\rangle
\end{bmatrix}$ should be positive semidefinite.
A: By the simple argument given by Jonas and explained in the Wiki page Theo referred to, there can exist no such positive measure $\mu$ since, using the notation below, $M_2M_4<(M_3)^2$. 
As regards the existence of a signed measure $\mu$ of bounded variation on the interval $[0,1]$ with a given sequence of moments 
$$
M_n=\displaystyle\int_0^1x^n\mathrm{d}\mu(x),
$$
it might be worthwhile, following Jonas's hint, to recall the result stated in chapter 3 section 2 of Widder's The Laplace transform: such a measure $\mu$ exists iff the sequence $(S_n)$ is bounded, where for every nonnegative $n$,
$$
S_n=\sum_{k=0}^n{n\choose k}\left|\sum_{i=0}^k(-1)^i{k\choose i}M_{n-i}\right|.
$$
Hence to determine if $(S_n)$ is bounded or not when $M_n=\mathrm{e}^{-n^2}$ for every $n\ge1$ would give the answer.
