# Are there any conditions on the moments that make a measure a probability measure?

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?

• Since there is a nonzero function on the real line with all moments zero, the answer is most likely no – Dirk Dec 24 '18 at 9:09
• 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 – Alon Amit Dec 24 '18 at 9:11
• Ah, since you also have that the density is positive, the answer may be yes... – Dirk Dec 24 '18 at 9:15
• @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. – maridia Dec 24 '18 at 9:34
• @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. – Alon Amit Dec 24 '18 at 9:40