Asymptotic properties of Azarin limit sets

Measure theory plays an important role in the theory of subharmonic and -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 limit sets in the space of Radon measures. The results of our article allow us to simplify the constructions from these articles somewhat.