Exercise 26, § 5

We denote by $x$ the identity mapping of $\mathbf{R}_+$.

a) Let $\nu$ be a positive measure on $\mathbf{R}_+$ such that $x^n$ is $\nu$-integrable for every integer $n\geqslant 0$. Put $\mu_n=\nu (x^n)$. For every integer $n\geqslant 0$ and for every complex numbers $a_0, . . .,a_n$, one has

b) If $(\mu_n)_{n\geqslant 0}$ is a sequence of real numbers satisfying the condition of the preceding question, then there exists a positive measure $\nu$ on $\mathbf{R}_+$ such that $\mu_n=\nu (x^n)$ for every $n\in \mathbf{N}$. ("Stieltjes moment problem"; use the method of Exercise 21, appealing to the Friedrichs extension.)