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?