Suppose i have a measure $\mu$ over $\mathbb R_+$ given by it's moments $\mu_0,...,\mu_n$, defined as :

$$\mu_k = \int x^{k} \partial\mu(x),\; k \in 1,...,n$$ Using Faa di bruno's formula, I can obtain the corresponding cumulants $\kappa_0,...,\kappa_n$.

Say that it exists another measure $\nu$ that happend to have the set of moments $\kappa_0,...,\kappa_n$.

Is there some work somewhere about the relationship between $\mu$ and $\nu$ ?

Suppose I have a measure $\mu$ over $\mathbb R_+$ given by its moments $\mu_0,...,\mu_n$, defined as :

$$\mu_k = \int x^{k} \partial\mu(x),\; k \in 1,...,n$$ Using Faà di Bruno's formula, I can obtain the corresponding cumulants $\kappa_0,...,\kappa_n$.

Say that there exists another measure $\nu$ that happens to have the set of moments $\kappa_0,...,\kappa_n$.

Is there some work somewhere about the relationship between $\mu$ and $\nu$ ?