Carleman's condition

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Carleman’s condition is a simple test to know when a set of moments comes from a unique measure.

What it says
- If a measure μ on the real line has finite moments m_n = ∫ x^n dμ(x) for all n, and the sum
∑_{n=1}^∞ (m_{2n})^{-1/(2n)}
diverges (goes to infinity), then μ is the only measure on the real line that has those moments. In other words, the moment problem is determinate.

Two common versions
- Hamburger moment problem (moments on the whole real line): the same condition with m_{2n} applies. If ∑_{n=1}^∞ m_{2n}^{-1/(2n)} = ∞, the moment sequence determines μ uniquely.
- Stieltjes moment problem (moments on [0, ∞)): the condition also guarantees determinacy when ∑_{n=1}^∞ m_n^{-1/(2n)} = ∞.

History
- Carleman, Torsten, discovered this criterion in 1922.

Caveat and generalization
- Later work showed that when the integrand is an arbitrary function, Carleman’s condition is not always enough to guarantee determinacy. A counter-example exists. To handle the broader case, a generalized Carleman condition has been proposed as a sufficient criterion for determinacy.