TITLE: When are probability distributions completely determined by their moments?
QUESTION [62 upvotes]: If two different probability distributions have identical moments, are they equal?  I suspect not, but I would guess they are "mostly" equal, for example, on everything but a set of measure zero.  Does anyone know an example of two different probability distributions with identical moments?  The less pathological the better.
Edit: Is it unconditionally true if I specialize to discrete distributions?
And a related question: Suppose I ask the same question about Renyi entropies.  Recall that the Renyi entropy is defined for all $a \geq 0$ by
$$ H_a(p) = \frac{\log \left( \sum_j p_j^a \right)}{1-a} $$
You can define $a = 0, 1, \infty$ by taking suitable limits of this formula.  Are two distributions with identical Renyi entropies (for all values of the parameter $a$) actually equal?  How "rigid" is this result?  If I allow two Renyi entropies of distributions $p$ and $q$ to differ by at most some small  $\epsilon$ independent of $a$, then can I put an upper bound on, say, $||p-q||_1$ in terms of $\epsilon$?  What can be said in the case of discrete distributions?

REPLY [3 votes]: I stumbled across this post while googling a question about moment determinacy. Then I also found this survey (Recent Developments on the Moment Problem, Gwo Dong Lin), which summarizes pretty much everything said here on the moment problem and gathers a lot more checkable conditions for moment (in)determinacy of probability distributions. I found it to be a great resource.