On a Class of Moment Problems

Consider a class M of probability measures on a measurable space X and measurable functions gi and h on X. In a typical moment problem we want further information on the possible sets of values taken by the integrals micrometers (gj) and micrometers (h) as micrometers runs through M. And the main purpose of the present paper is to develop into more systematic methods certain principles which in special cases have been found effective for handling such moment problems. In Sections 2 through 4 we take up certain frequently occurring moment problems where the class X happens to be convex. In Section 2 the space X can be any locally compact Hausdorff space. For {hj, j epsilon J} as an arbitrary collection (finite or infinite) of lower semicontinuous functions on X, we establish a condition which is both necessary and sufficient for the existence of a regular probability measure micrometer on X satisfying y (hj) less than or equal to nj for all j epsilon J. However, we do assume as a side condition that the hj dominate each other at infinity in a certain weak sense. This domination condition is void when X is compact and nearly so when hj equal to or more than 0.