The Generalized Moment Problem Associated with Correlation Measures

The classical power moment problem can be viewed as a theory of spectral representations of a positive functional on some classical commutative algebra with involution. We generalize this approach to the case in which the algebra is a special commutative algebra of functions on the space of multiple finite configurations. If the above-mentioned functional is generated by a measure on the space of finite ordinary configurations, then this measure is the correlation measure for a measure on the space of infinite configurations. The positiveness of the functional gives conditions for the measure to be a correlation measure.