Hausdorff moment problem

Remarks

The problem is solved. Write $(\Delta m)_n = m_{n+1} - m_n$ for the difference operator. Hausdorff proved that such a measure exists if and only if the sequence is completely monotone, that is, $$(-1)^k (\Delta^k m)_n \ge 0 \quad \text{for all } n, k \ge 0.$$ When the measure exists, it is unique. Uniqueness holds because polynomials are dense in the continuous functions on $[0, 1]$ by the Weierstrass approximation theorem. For example, $m_n = 1/(n + 1)$ is the moment sequence of Lebesgue measure on $[0, 1]$.