Indeterminate moment problem

Remarks

Example: the log-normal density $f(x) = (x \sqrt{2\pi})^{-1} e^{-(\ln x)^2/2}$ on $(0, \infty)$. The substitution $x = e^{u}$ shows $\int_0^\infty x^n f(x) \sin(2\pi \ln x) \, dx = e^{n^2/2} \int_{\R} \frac{e^{-v^2/2}}{\sqrt{2\pi}} \sin(2\pi v) \, dv = 0$, with $v = u - n$, since the last integrand is odd. So $\nu$ with density $f(x)(1 + \sin(2\pi \ln x))$ is a different positive measure on $[0, \infty)$ with the same moments. Non-example: the uniform distribution on $[0, 1]$ is not indeterminate, since a measure on a compact set is determined by its moments. Every measure with finitely many atoms is determinate, so an indeterminate measure has infinite support; its support must also be unbounded.