# Thinness of a set

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff) at a point A local criterion for the fact that is a polar set. A non-empty set is said to be thin at the point in two cases:

1) is not a limit point of ; that is, , where is the derived set of ;

2) and there exists a superharmonic function in a neighbourhood of (see Superharmonic function) such that The set is polar if and only if it is thin at each of its points. For an arbitrary set the subset of those points at which is thin is polar. Any non-empty subset of a set which is thin at the point is thin at . The union of a finite number of sets thin at the point is a set thin at .

A segment in the plane is not a thin set at any of its points. If is a thin set at a point , then there exist arbitrarily small discs with centre and not intersecting . A polar set is completely discontinuous. However, the Cantor set on the -axis (which is of measure zero) is not thin at any of its points. At the same time, for example, in the set of points which has a spine at the point , where is the Newton potential with density on the segment , is a thin set at the spine (Lebesgue's example).

How to Cite This Entry:
Thinness of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thinness_of_a_set&oldid=18916
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article