I am interested in criteria that guarantee that a Borel probability measure has compact support. I outline two below and I am hoping to gather more as answers (if they exist).
The first sufficient condition I know for a Borel probability measure $\mu$ (say) on $\mathbb{R}$ which has all moments of order $k$ for any $k\ge 1$ is as follows. If there exists $C, R>0$ such that for any $k\ge 1$, $$\int |x|^k d\mu(x)\le C R^k,$$ then $$\operatorname{supp}\mu \subseteq \{|x|\le R\}.$$ The proof uses Chebyshev's inequality. Let $\lambda>R$. Then, $$\mu\left\{x\in \mathbb{R}:|x|>\lambda\right\}\le \frac{1}{\lambda^k}\int |x|^kd\mu(x)\le C \left(\frac{R}{\lambda}\right)^k\to 0$$ if we let $k\to\infty.$ Of course, the converse also holds.
The second condition is the Paley-Wiener result given in this answer.
Unlike in the linked answers above, I am interested in merely sufficient conditions guaranteeing the compactness of the support with the hope that there are more results of this kind in addition to the ones I quoted above.
