The problem of moments, also known as moment problem, is the elegant question of establishing whether there exists a probability measure having some prescribed moments. A fascinating feature of this problem is its great modeling power, which has been attracting scientists for more than a century in the most different scientific areas. This talk aims to give an introduction to the moment problem with a special focus on the connection between finite and infinite dimensional moment problem. In fact, the theory of moment problems for probabilities supported in an infinite dimensional space is not as well developed as in the finite dimensional case and is still not up to the demand of the applications. We present a general approach to the infinite dimensional moment problem based on projective limit techniques which allow us to exploit the results in the classical finite dimensional moment theory to the infinite dimensional case. We prove that under the so-called Prokhorov condition, the infinite dimensional moment problem on a unital commutative algebra A is solvable if and only if for any finitely generated subalgebra S of A the corresponding finite dimensional moment problem is solvable. Among the several applications, we point out the case when A is the tensor algebra of an infinite dimensional vector space and discuss some consequences for moment problems for random measures.

This is a joint work with Salma Kuhlmann, Tobias Kuna and Patrick Michalski.