Thinness and Hyperthinness

Abstract

Concepts of “small” or in some sense exceptional sets play essential roles in many parts of analysis, probability theory as well as in both classical and probabilistic potential theory. There are many examples: sets of first category, sets of measure zero, polar sets, are all examples of such “small” sets. In many situations one has to compare these sets: although a set of first category is small and so is a set of measure zero, these sets are not the same. In potential theory a concept of “smallness” of a set at a point is of special interest. More precisely, the concept of a set thin at a point is of major interest in potential theory. Originally, this notion arose in classical potential theory in conjuction with Dirichlet problem. Here, the ultimate characterization of regularity is a necessary and sufficient condition due to N. Wiener [7]. THe so-called Wiener’s test gives necessary and sufficient conditions in terms of capacity for a point of a set to be irregular, i.e., for a set to be thin at a point. (See more in [3], [4], [5], [6]).