I want to prove that a positive measure $\mu$ on $\mathbb{R}$ with compact support is uniquely determined by its moments $$ m_k = \int_{\mathbb{R}}x^k d\mu(x). $$
I have looked through a lot of literature, but I only come across sources that say it's automatic or obvious.
The only tip I could find was from wikipedia's Moment Problem article: https://en.wikipedia.org/wiki/Moment_problem
The uniqueness of μ in the Hausdorff moment problem follows from the Weierstrass approximation theorem, which states that polynomials are dense under the uniform norm in the space of continuous functions on [0, 1].
But I don't know what to do with this information. We don't know that $\mu$ is continuous right?
So far this is what I have. If we have two compactly supported measures $\mu$ and $\nu$ with the same moment sequence $(m_n)_{n\geq 0}$, than they will have the same Laplace transform.
$$ \mathcal{L}[\mu](z) = \int_{\mathbb{R}}e^{-zx}d\mu(x) = \int_{-T}^T e^{-zx}d\mu(x) = \sum_{n\geq 0} \frac{(-z)^n}{n!}m_n = \mathcal{L}[\nu](z). $$
In addition we can show that $\exists C>0:\,$ $|m_n|\leq C^n$ because of the compact support. From this it follows that the Laplace transform is entire. If we can now invert the Laplace transform we would be done. But I'm not sure how this works, because the ordinary inverse Laplace transform that we can find e.g. here https://en.wikipedia.org/wiki/Inverse_Laplace_transform gives you a function whereas I would like a map from Borel subsets to $\mathbb{R}$.
In Summary: Can anyone help me prove uniqueness for the Hausdorff moment problem? Either along the way I was trying or any other. Thanks in advance.
A simple proof is based on Markov-Riesz representation theorem, which state that the dual of the space of continuous functions over a compact set is the set of radon measures of finite variation on the same compact : $$ (\mathcal{C}^0(K))^*=\mathcal{M}(K) $$ We consider the Riesz Functional : $$ \matrix{ \varphi : &\mathbb C[X]&\to & \mathbb C\\ & \sum_{k\le p} a_k x^k & \mapsto & \sum_{k\le p} a_k m_k} $$ Since the support is compact, we get that $$ |\varphi(p)|=\left|\int_K p(t)d\mu(t) \right|\le \|p\|_{\infty}\mu(K) $$ We can extend uniquely now (Thanks to the last inequality) the functional $\varphi$ to $\overline{\mathbb C[X]|_K}^\infty$, but by Stone-Weierstrass theorem $\overline{\mathbb C[X]}^\infty=\mathcal C^0(K)$.
So from the moment sequence $(m_k)_k$ we can construct a unique continuous linear functional $\varphi\in (\mathcal{C}^0(K))^*$, then by the uniqueness of the representing measure on Markov-Risez theorem, this measure will be unique solution of the moment problem.