The Stieltjes moment problem consists in finding the conditions on a sequence $m_n$ such that $$m_n = \int_0^\infty x^n d\mu(x)$$ for some positive measure $\mu$.
I am looking for a proof of this statement.
This seems to be mentioned between pp 1 and 2 of these pdf notes, using continuous fractions, but the explanation is not very explicit and I don't quite understand the connection between continuous fractions and the problem at hand. A note to this problem is also made in Simon and Reed's Methods of Modern Mathematical Physics (1972) (pag 145), but only in the form of an exercise to the reader. I am aware that I could look up the original 1894 paper but I was hoping for a more recent exposition/proof.