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.

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_a(p) = log(Σ_j p_j^a)/(1-a)

You can define a=0,1,∞ 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 ε independent of a, then can I put an upper bound on, say, || p - q ||₁ in terms of ε?