Thinness of a set

$ E \subset \mathbf R ^ {n} $ at a point $ y _ {0} \in \mathbf R ^ {n} $

A local criterion for the fact that $ E $ is a polar set. A non-empty set $ E \subset \mathbf R ^ {n} $ is said to be thin at the point $ y _ {0} \in \mathbf R ^ {n} $ in two cases:

1) $ y _ {0} $ is not a limit point of $ E $; that is, $ y _ {0} \notin E ^ \prime $, where $ E ^ \prime $ is the derived set of $ E $;

2) $ y _ {0} \in E ^ \prime $ and there exists a superharmonic function $ v ( x) $ in a neighbourhood of $ y _ {0} $( see Superharmonic function) such that

$$ \lim\limits _ {\begin{array}{c} x \rightarrow y _ {0} \\ x \in E \setminus \{ y _ {0} \} \end{array} } \ \inf v ( x) > v ( y _ {0} ). $$

The set $ E $ is polar if and only if it is thin at each of its points. For an arbitrary set $ E $ the subset of those points at which $ E $ is thin is polar. Any non-empty subset of a set which is thin at the point $ y _ {0} \in \mathbf R ^ {n} $ is thin at $ y _ {0} $. The union of a finite number of sets thin at the point $ y _ {0} \in \mathbf R ^ {n} $ is a set thin at $ y _ {0} $.

A segment in the plane $ \mathbf R ^ {2} $ is not a thin set at any of its points. If $ E \subset \mathbf R ^ {2} $ is a thin set at a point $ y _ {0} $, then there exist arbitrarily small discs with centre $ y _ {0} $ and not intersecting $ E $. A polar set $ E \subset \mathbf R ^ {2} $ is completely discontinuous. However, the Cantor set on the $ x $- axis (which is of measure zero) is not thin at any of its points. At the same time, for example, in $ \mathbf R ^ {3} $ the set of points

$$ E = \{ {( x, y, z) } : {V ( x, y, z) \geq k > 1 } \} , $$

which has a spine at the point $ ( 0, 0, 0) $, where

$$ V ( x, y, z) = \ \int\limits _ { 0 } ^ { 1 } \frac{t dt }{\sqrt {( x - t) ^ {2} + y ^ {2} + z ^ {2} } } $$

is the Newton potential with density $ t $ on the segment $ ( 0 \leq x \leq 1, 0, 0) $, is a thin set at the spine $ ( 0, 0, 0) \in E ^ \prime $( Lebesgue's example).

References

[1]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1969)
[2]	N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)

Comments

Two further interesting properties of thinness are: 1) $ E $ is thin at $ x $ if and only if $ x $ is not a limit point of $ E $ with respect to the fine topology; and 2) a boundary point $ x $ of an open set $ U $, bounded if $ U \subset \mathbf R ^ {2} $, is regular for the Dirichlet problem if and only if the complement of $ U $ is not thin at $ x $.

The concept of thinness, and its use to define a fine topology, is fundamental in any potential theory. For example, in probabilistic potential theory associated to a strong Markov process, a Borel set $ E $ is thin at $ x $ if and only if, starting from $ x $, the process almost surely will not hit $ E $ even once. But, in general, a set thin at each of its points is not polar; a countable union of such sets is called a semi-polar set, a kind of exceptional set (related to the Dirichlet problem) which can be considerably bigger than a polar set when the potential theory lacks symmetry (for example, for the heat equation potential theory). Roughly speaking, a set $ E $ is polar (respectively semi-polar) in probabilistic potential theory if the process almost surely never meets $ E $( respectively, only meets $ E $ at most a countable number of times). See also Potential theory, abstract.

References

[a1]	C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)