The Riesz–Radon Problem of Characterizing Integrals and the Weak Compactness of Radon Measures

The problem of characterizing integrals considered in this paper dates back to the fundamental works of Riesz (1909), Radon (1913), and Frechet 1914). A solution to this problem is given in the form of a general parametric theorem, which implies the following theorems as particular cases: (1) the Riesz–Radon theorem for a locally compact space, (2) the Prokhorov theorem for a Tikhonov space, and (3) an integral representation theorem for an arbitrary Hausdorff space. A weak compactness criterion for the sets of bounded Radon measures on an arbitrary Hausdorff space is derived as an application of the last theorem. This criterion dates back to the Prokhorov criterion for a Polish space and to the Prokhorov–Le Cam theorem for a Tikhonov space.