Krein's condition

Remarks

The integral is well defined in $(-\infty, \infty]$, since $\ln f \le f$ where $f \ge 1$ and $f$ is integrable. The result is a corollary of a density theorem of Krein: the integral is infinite if and only if the exponential sums $\sum_k a_k e^{i \lambda_k x}$ with $\lambda_k \ge 0$ are dense in $L^2(\mu)$. Example: $f(x) = c \, e^{-\abs{x}^{\alpha}}$ with $0 < \alpha < 1$ and $c > 0$ has $-\ln f(x) = \abs{x}^{\alpha} - \ln c$, and $\int \abs{x}^{\alpha}/(1 + x^2) \, dx < \infty$ for $\alpha < 1$, so this density is not determined by its moments. For $\alpha \ge 1$ the integral diverges, and in fact Carleman's condition shows that the moment problem is determinate. The condition is sufficient for indeterminacy but not necessary.

Going deeper

For a student

It complements Carleman's condition, which gives determinacy when the moments grow slowly. The two meet at densities like $e^{-\abs{x}^{\alpha}}$: determinate for $\alpha \ge 1$ and indeterminate for $\alpha < 1$. Versions for the Stieltjes problem on $[0, \infty)$ and converses under regularity hypotheses on $f$ are known.