Asymptotic characteristics of measures, limit Azarin sets

Azarin's theory of limit sets plays an important role in the theory of subharmonic and 8 -subharmonic functions. The classical properties of a measure have been presented in many monographs, for example, in [1]. In the article we sharpen Azarin's variant of the theorem on a compact set in the space of Radon measures. The results of our article allow us to simplify the constructions from these articles somewhat.