Determinate moment problem

Remarks

Example: the standard normal distribution is determinate on $K = \R$. Its moments are $m_{2n} = (2n - 1)!! \le (2n)^n$ and $m_{2n+1} = 0$, so $\sum_n m_{2n}^{-1/(2n)}$ diverges, and Carleman's condition applies. Example: every measure with support in a compact set $K$ is determinate on $K$, by the density of polynomials in $C(K)$. Non-example: the log-normal distribution with density $f(x) = (x \sqrt{2\pi})^{-1} e^{-(\ln x)^2/2}$ on $(0, \infty)$ is not determinate, since $f(x)(1 + \sin(2\pi \ln x))$ is a different density with the same moments. Determinacy depends on $K$: a measure on $[0, \infty)$ can be determinate among measures on $[0, \infty)$ (Stieltjes determinate) but not among measures on $\R$ (Hamburger determinate).