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?

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    $\begingroup$ Since there is a nonzero function on the real line with all moments zero, the answer is most likely no $\endgroup$ – Dirk Dec 24 '18 at 9:09
  • $\begingroup$ This seems to be a special case of the Markov Moment Problem, where you include the zeroth moment $m_0$ to ensure that the total measure is $1$. Is this what you are looking for? en.wikipedia.org/wiki/Moment_problem $\endgroup$ – Alon Amit Dec 24 '18 at 9:11
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    $\begingroup$ Ah, since you also have that the density is positive, the answer may be yes... $\endgroup$ – Dirk Dec 24 '18 at 9:15
  • $\begingroup$ @AlonAmit, I don't think that wikipedia article references the Markov problem (I found a statement anyhow). Yes, I think the Markov Moment Problem gives a partial answer. It seems to be restricted to measures of bounded support though. $\endgroup$ – maridia Dec 24 '18 at 9:34
  • $\begingroup$ @maridia no, the moment problem covers the unbounded support case as well. It is sometimes called more specifically the en.wikipedia.org/wiki/Hamburger_moment_problem. $\endgroup$ – Alon Amit Dec 24 '18 at 9:40

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