For a positive Borel measure $\mu$ on the ~~ real line ~~ interval $[-1, 1]$, let $\displaystyle{m_n = \int_{-\infty}^\infty x^n d\mu(x)}$, i.e. the $n$th moments of the measure. Are there any conditions on $m_n$, $n \ge 1$, for when $\mu$ can be made into a probability measure, i.e. $\mu(\mathbb{R}) < \infty$? Recall that the moments of a uniform distribution on $[-1,1]$ are $1/(n+1)$ for even $n$. The measure given by density $p(x)= 1/(2|x|)$ for $x \in [-1,1]$ has moments $1/n$ for even $n$. So it seems that two moment sequences with very similar asymptotic behavior can give different answers to this question.

**Edit:**
Actually, consider the additional example of the densities $1/(4\sqrt{x})$ and $1/4(1/x-1/\sqrt{x})$ in $[-1,1]$ which have even moments $1/(2n+1)$ and $1/(2n(2n+1))$ respectively. So some sort of upper bound condition on the moments is certainly out of the question. In fact, are there two measures $\mu$ with and $\nu$ that both have the moment sequence $m_1,m_2,...$ where one is finite and the other is infinite?