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` ≥ 0 by

H<sub>a</sub>(p) = log(&sum;<sub>j</sub> p<sub>j</sub><sup>a</sup>)/(1-a)

You can define `a`=0,1,&infin; 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 ||<sub>1</sub> in terms of &epsilon;?  What can be said in the case of discrete distributions?